The seminar covers a wide range of topics in Pure and Applied Probability, Mathematical Physics, Ergodic Theory and Dynamical Systems. The seminar is organized by Ron Peled.

Usual place and time are Tel Aviv University's Schreiber Building room 309, on Mondays at 14:30-15:30, but check each announcement since this is sometimes changed.

The seminar is named after Shlomo Horowitz, who was among the first faculty members to study probability theory at Tel Aviv University.

Monday, March 5

Erwin Bolthausen, Universität Zürich

On the Pekar process and its connection with the polaron problem (joint work with Wolfgang Koenig and Chiranjib Mukherjee)

**Abstract:**

On the Pekar process and its connection with the polaron problem (joint work with Wolfgang Koenig and Chiranjib Mukherjee)

A long standing open problem is the asymptotics of the effective mass
for the Froehlich polaron in the strong coupling limit. Already Feynman formulated
the problem in terms of path integrals, which leads to a three dimensional Brownian motion with
an attractive path interaction with a singular interaction kernel.
In the celebrated paper by Donsker and Varadhan (Comm Pure Appl Math, 1983), the
asymptotics of the free energy is considered. The effective mass is however closely
tied to the path behavior which is more delicate. In a heuristic derivation by Spohn
(Phys Rev B, 1986) the problem is related to the behavior of a stochastic process, Spohn
called the "Pekar process". The problem about the asymptotics of the effective mass
is mathematically still open. In the work with Koenig and Mukherjee (Comm Pure Appl Math 2017),
we construct rigorously the Pekar process as the asymptotic limit of the Brownian motion
with the singular pair interaction, and we discuss the conjectured relation with the original problem.

Schreiber Building Room 309 at 14:30. Monday, March 12

Ariel Yadin, Ben-Gurion University

Harmonic functions on topological groups

**Abstract:**

Harmonic functions on topological groups

We will discuss recent results regarding the relation between growth and harmonic functions on topological groups.
Specifically, the use of random walks and probabilistic tools to obtain geometric results, and relate the algebraic and geometric structures.

Schreiber Building Room 309 at 14:30. Monday, March 19

Santiago Saglietti, Technion

A strong law of large numbers for supercritical BBM with absorption

**Abstract:**

A strong law of large numbers for supercritical BBM with absorption

We consider a (one-dimensional) branching Brownian motion process with a general offspring distribution having at least two finite moments, in which all particles have a drift towards the origin and are immediately killed if they reach it. It is well-known that if and only if the branching rate is sufficiently large, the population survives forever with a positive probability. We show that throughout this super-critical regime, the number of particles inside any given set normalized by the mean population size converges to an explicit limit, almost surely and in L^{1}. As a consequence, we get that, almost surely on the event of survival of the branching process, the empirical distribution of particles converges weakly to the (minimal) quasi-stationary distribution associated with the Markovian motion driving the particles. This proves a result of Kesten from 1978, for which no proof was available until now. Joint work with Oren Louidor.

Schreiber Building Room 309 at 14:30. Monday, March 26 and Monday, April 2

No seminar - university is on a Wednesday schedule and Passover break.Monday, April 9

Benny Sudakov, ETH Zürich

Asymptotic in bond percolation on expanders

**Abstract:**

Asymptotic in bond percolation on expanders

The evolution of the largest connected component has been studied intensely
in a variety of random graph processes, starting with celebrated work of
Erdos-Renyi, who in 1960 proved that a random subgraph of an n-vertex
complete graph undergoes a phase transition at edge probability 1/n when,
asymptotically almost surely, a linear-sized (``giant'') component appears.

In this talk we consider edge percolation on a family of high-girth d-regular expanders. Alon-Benjamini-Stacey in 2004 established that a critical probability for the appearance of a giant component in this case is p_{c}=1/(d-1). Our main result recovers the sharp asymptotics, similar to the
classical Erdos-Renyi result, of the size and degree distribution of the
vertices in the giant component at any p>p_{c}. On the other hand we show that,
unlike the situation in the classical random graph case, the second largest
component in edge percolation on a regular expander, even with an arbitrarily
large girth, can have size at least n^{a} for any fixed a<1. Several related
results will be discussed as well.

Joint work with Michael Krivelevich and Eyal Lubetzky.

Schreiber Building Room 309 at 14:30. In this talk we consider edge percolation on a family of high-girth d-regular expanders. Alon-Benjamini-Stacey in 2004 established that a critical probability for the appearance of a giant component in this case is p

Joint work with Michael Krivelevich and Eyal Lubetzky.

Monday, April 16

Ron Rosenthal, Technion

Stabilization of Diffusion Limited Aggregation in a Wedge

**Abstract:**

Stabilization of Diffusion Limited Aggregation in a Wedge

We prove a discrete Beurling estimate for the harmonic measure in a wedge in Z^{2}, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than π/4 stabilizes.
This allows to consider the infinite DLA and ask questions about its growth, dimension and number of arms. Some conjectures and open problems will be discussed.

Based on a joint work with Eviatar Procaccia and Yuan Zhang.

Schreiber Building Room 309 at 14:30. Based on a joint work with Eviatar Procaccia and Yuan Zhang.

Monday, April 23

Mark Rudelson, University of Michigan

Invertibility of the adjacency matrices of random graphs

**Abstract:**

Invertibility of the adjacency matrices of random graphs

Consider an adjacency matrix of a bipartite, directed, or undirected Erdos-Renyi random graph. If the average degree of a vertex is large enough, then such matrix is invertible with high probability. As the average degree decreases, the probability of the matrix being singular increases, and for a sufficiently small average degree, it becomes singular with probability close to 1. We will discuss when this transition occurs, and what the main reason for the singularity of the adjacency matrix is.

This is a joint work with Anirban Basak.

Schreiber Building Room 309 at 14:30. This is a joint work with Anirban Basak.

Monday, April 30

Qingsan Zhu, Tel Aviv University

TBA

Schreiber Building Room 309 at 14:30.

TBA

Schreiber Building Room 309 at 14:30.

Monday, May 7

TBA

Schreiber Building Room 309 at 14:30.

Schreiber Building Room 309 at 14:30.

Monday, May 14

Gady Kozma, Weizmann Institute

TBA

Schreiber Building Room 309 at 14:30.

TBA

Schreiber Building Room 309 at 14:30.

Monday, May 21

TBA

Schreiber Building Room 309 at 14:30.

Schreiber Building Room 309 at 14:30.

Monday, May 28

Yotam Smilansky, Tel Aviv University

TBA

Schreiber Building Room 309 at 14:30.

TBA

Schreiber Building Room 309 at 14:30.

Monday, June 4

Oren Louidor, Technion

TBA

Schreiber Building Room 309 at 14:30.

TBA

Schreiber Building Room 309 at 14:30.

Monday, June 11

TBA

Schreiber Building Room 309 at 14:30.

Schreiber Building Room 309 at 14:30.

Monday, June 18

Special after semester seminar!
Tyler Helmuth, University of Bristol

TBA

Schreiber Building Room 309 at 14:30.

TBA

Schreiber Building Room 309 at 14:30.

Monday, October 23

Daniel Jerison, Tel Aviv University

Random walks on sandpile groups

**Abstract:**

Random walks on sandpile groups

The sandpile group of a finite graph is an Abelian group that is defined using the graph Laplacian. I will describe a natural random walk on this group. The main questions are: how long does it take for the sandpile random walk to mix, and how is the mixing time related to the geometry of the underlying graph? These questions can sometimes be answered even if the actual group is unknown. In particular, the spectral gaps of the sandpile walk and of the simple random walk on the underlying graph exhibit a surprising inverse relationship. In certain cases, the sandpile walk exhibits "cutoff" behavior: the Markov chain goes from almost completely unmixed to almost completely mixed in a relatively short number of steps. I will give a tour of what we know about the sandpile walk and briefly discuss how some of the results are proved. This is joint work with Bob Hough, Lionel Levine, and John Pike.

Schreiber Building Room 309 at 14:30. Monday, October 30

Matan Harel, Tel Aviv University

Discontinuity of the phase transition for the planar random-cluster and Potts models with q > 4

**Abstract:**

Discontinuity of the phase transition for the planar random-cluster and Potts models with q > 4

The ferromagnetic q-Potts Model is a classical spin system in which one of q colors is placed at every vertex of a graph and the configuration is assigned an energy proportional to the number of monochromatic neighbors. It is highly related to the random-cluster model, which is a dependent percolation model where a configuration is weighted by q to the power of the number of connected components. Through non-rigorous means, Baxter showed that the phase transition is first-order whenever q > 4 - i.e. there exist multiple Gibbs states at criticality. We provide a rigorous proof of this claim. Like Baxter, our proof uses the correspondence between the above models and the six-vertex model, which we analyze using the Bethe ansatz and transfer matrix techniques. We also prove Baxter's formula for the correlation length of the models at criticality. This is joint work with Hugo Duminil-Copin, Maxime Gangebin, Ioan Manolescu, and Vincent Tassion.

Schreiber Building Room 309 at 14:30. Monday, November 6

Raimundo Briceño, Tel Aviv University

Combinatorial aspects of entropy computation and factoring in multidimensional symbolic dynamics

**Abstract:**

Combinatorial aspects of entropy computation and factoring in multidimensional symbolic dynamics

Finite-valued Markov random fields with hard constraints and some Gibbsian distribution appear in several contexts. They are naturally related to statistical physics (e.g., proper q-colorings), combinatorics (e.g., graph homomorphisms), computer science (e.g., counting independent sets), among other areas. In this talk we take a symbolic dynamical approach. For concreteness, we focus on systems where the graph is the Z^{d} lattice and consider two intimately related problems: entropy/pressure computation and (topological) factoring of shift spaces. We will show how a combinatorial version of the measure-theoretic property known as strong spatial mixing has been useful for creating a framework to generalize many previous results. Time permitting, we relate this approach with work of Brightwell and Winkler on the characterization of hard constraints represented by dismantlable graphs. Most of the entropy computation and factoring results are based on joint works with Adams, Marcus, McGoff, and Pavlov; the results on dismantlablility are based on ongoing joint work with Bulatov, Dalmau, and Larose.

Schreiber Building Room 309 at 14:30. Monday, November 13

Anirban Basak, Weizmann Institute

Invertibility and condition number of sparse random matrices

**Abstract:**

Invertibility and condition number of sparse random matrices

The condition number, i.e. the ratio of the largest and the smallest singular value, of a matrix serves as a measure of stability for matrix algorithms. Based on simulations, von Neumann and his collaborators conjectured that the condition number of a random square matrix of dimension n is O(n). During the last decade, this conjecture was proved for dense random matrices. In this talk I will consider sparse random matrices such as matrices with i.i.d. Ber(p_n) entries where p_n tends to 0 as n tends to infinity. I will describe our work that establishes (akin to) von Neumann's conjecture on condition number for sparse random matrices. This talk is based on joint works with Mark Rudelson.

Schreiber Building Room 309 at 14:30. Monday, November 20

Ofer Zeitouni, Weizmann Institute

On the cover time of the two-dimensional sphere by Brownian motion

**Abstract:**

On the cover time of the two-dimensional sphere by Brownian motion

Let C denote the cover time of the two-dimensional sphere by a Wiener sausage of radius r. We prove that sqrt(C) - 2sqrt(2)(log(1/r) - 1/4 log(log(1/r))) is tight.
I will describe the background and explain why dimension 2 is special for this question.

This is joint work with David Belius and Jay Rosen.

Schreiber Building Room 309 at 14:30. This is joint work with David Belius and Jay Rosen.

Monday, November 27

Mauro Artigiani, Scuola Normale Superiore

Hall rays for Lagrange spectra at cusps of Riemann surfaces

**Abstract:**

Hall rays for Lagrange spectra at cusps of Riemann surfaces

The Lagrange spectrum is a classical object in Diophantine approximation on the real line. It can be also seen as the spectrum of asymptotic penetration of hyperbolic geodesics into the cusp of the modular surface. This interpretation yielded many generalizations of the Spectrum to non-compact, finite volume, negatively curved surfaces and higher dimensional manifolds. A remarkable property of the classical Spectrum is that it contains an infinite interval, called Hall ray. The presence of the Hall ray is a common feature of the generalizations of the Lagrange spectrum to higher dimensions. We show that the Lagrange spectrum of hyperbolic surfaces contains a Hall ray. Moreover, we show that the same result holds if we measure the excursion into the cusps with a proper function that is close in the Lipschitz norm to the hyperbolic height.

This is a joint work with L. Marchese and C. Ulcigrai.

Schreiber Building Room 309 at 14:30. This is a joint work with L. Marchese and C. Ulcigrai.

Monday, December 4

Ewain Gwynne, MIT

A mating-of-trees approach for graph distances and random walk on random planar maps

**Abstract:**

A mating-of-trees approach for graph distances and random walk on random planar maps

We discuss a general strategy for proving estimates for a certain family of random planar maps. The random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a ``mating-of-trees" type bijection, and includes the uniform infinite planar triangulation (UIPT) and the infinite-volume limits of random planar maps sampled with probability proportional to the number of spanning trees, bipolar orientations, or Schnyder woods they admit.

Using this strategy, we obtain non-trivial estimates for graph distances in natural non-uniform random planar maps. We also prove that the spectral dimension of a class of random planar maps (including the UIPT) is a.s. equal to 2 - i.e., the return probability to the starting point after n steps is n^{-1+o(1)} - and obtain the correct lower bound of n^{1/4 - o(1)} for the graph-distance displacement of the simple random walk on the UIPT.

Our approach proceeds by way of a strong coupling of the encoding walk for the map with a correlated two-dimensional Brownian motion (Zaitsev, 1998), which allows us to compare our given map with the so-called mated-CRT map constructed from this correlated two-dimensional Brownian. The mated-CRT map is closely related to SLE-decorated Liouville quantum gravity due to results of Duplantier, Miller, and Sheffield (2014). So, we can analyze the mated-CRT map using continuum theory and then transfer to other random planar maps via strong coupling. We expect that this approach will have further applications in the future.

Based on joint works with Nina Holden and Xin Sun, and with Jason Miller. arxiv.org/abs/1711.00723, arxiv.org/abs/1711.00836

Schreiber Building Room 309 at 14:30. Using this strategy, we obtain non-trivial estimates for graph distances in natural non-uniform random planar maps. We also prove that the spectral dimension of a class of random planar maps (including the UIPT) is a.s. equal to 2 - i.e., the return probability to the starting point after n steps is n

Our approach proceeds by way of a strong coupling of the encoding walk for the map with a correlated two-dimensional Brownian motion (Zaitsev, 1998), which allows us to compare our given map with the so-called mated-CRT map constructed from this correlated two-dimensional Brownian. The mated-CRT map is closely related to SLE-decorated Liouville quantum gravity due to results of Duplantier, Miller, and Sheffield (2014). So, we can analyze the mated-CRT map using continuum theory and then transfer to other random planar maps via strong coupling. We expect that this approach will have further applications in the future.

Based on joint works with Nina Holden and Xin Sun, and with Jason Miller. arxiv.org/abs/1711.00723, arxiv.org/abs/1711.00836

Monday, December 11

Nattalie Tamam, Tel Aviv University

Divergent trajectories in arithmetic homogeneous spaces of rational rank two

**Abstract:**

Divergent trajectories in arithmetic homogeneous spaces of rational rank two

In the theory of Diophantine approximations, singular points are ones for which Dirichlet’s theorem can be infinitely improved. It is easy to see that all rational points are singular. In the special case of dimension one, the only singular points are the rational ones. In higher dimensions, points lying on a rational hyperplane are also obviously singular. However, in this case there are additional singular points. In the dynamical setting the singular points are related to divergent trajectories. In the talk I will define obvious divergent trajectories and explain the relation to rational points. In addition, I will present the more general setting involving Q-algebraic groups. Lastly I will discuss results concerning classification of divergent trajectories in Q-algebraic groups.

Schreiber Building Room 309 at 14:30. Monday, December 18

Amir Dembo, Stanford University

Walking within growing domains: recurrence versus transience

**Abstract:**

Walking within growing domains: recurrence versus transience

When is simple random walk on growing in time d-dimensional domains recurrent? For domain growth which is independent of the walk, we review recent progress and related universality conjectures about a sharp recurrence versus transience criterion in terms of the growth rate. We compare this with the question of recurrence/transience for time varying conductance models, where Gaussian heat kernel estimates and evolving sets play an important role. We also briefly contrast such expected universality with examples of the rich behavior encountered when monotone interaction enforces the growth as a result of visits by the walk to the current domain’s boundary.

This talk is based on joint works with Ruojun Huang, Vladas Sidoravicius and Tianyi Zheng.

Schreiber Building Room 309 at 14:30. This talk is based on joint works with Ruojun Huang, Vladas Sidoravicius and Tianyi Zheng.

Monday, December 25

Scott Aaronson, University of Texas at Austin

BosonSampling and the Permanents of Gaussian Matrices

**Abstract:**

BosonSampling and the Permanents of Gaussian Matrices

I'll discuss random matrix theory questions that arise in
BosonSampling: a 2011 proposal by myself and Alex Arkhipov to
demonstrate "quantum computational supremacy" (that is, an exponential
speedup over classical computers) using a rudimentary quantum optical
setup. The goal, in BosonSampling, is to sample from a certain kind
of probability distribution, in which the probabilities are given by
the absolute squares of permanents of complex matrices (n-by-n
matrices, if there are n photons involved). Of particular interest to
probabilists is that the BosonSampling program leads naturally to rich
mathematical questions about the permanents of i.i.d. Gaussian
matrices. For example: are such permanents close to lognormally
distributed? Is there an efficient algorithm to estimate these
permanents, for most matrices? I'll survey what we know and don't
know about these questions, and invite the audience to help solve
them. No quantum computation background is needed for the talk.

Schreiber Building Room 309 at 14:30. Monday, January 1

Diana Davis, Swarthmore College

Tiling billiards and interval exchange transformations

**Abstract:**
*Tiling billiards* is a new dynamical system where a beam of light refracts through a planar tiling. It turns out that, for a regular tiling of the plane by congruent triangles, the light trajectories can be described by interval exchange transformations. I will explain this surprising correspondence, and show that we get the Rauzy fractal as a billiard trajectory.
Schreiber Building Room 309 at 14:30.

Tiling billiards and interval exchange transformations

Monday, January 8

Dor Elboim, Tel Aviv University

The cycle structure of random Euclidean Permutations

**Abstract:**

The cycle structure of random Euclidean Permutations

In the model of random Euclidean permutations one samples N points in a box in R^{d} and a permutation of the points, with a probability distribution which gives higher weight to configurations with short distances between consecutive points along the permutation. The object of study is the cycle structure of the permutation as the number of points and the size of the box tend to infinity while keeping the density of points controlled. Following Matsubara and Feynman, the emergence of macroscopic cycles in the permutation has been related to the phenomenon of Bose-Einstein condensation. For each dimension d≥1, we identify sub-critical, critical and super-critical regimes in terms of the density and find the limiting distribution of cycle lengths in each regime. The results extend earlier work of Betz and Ueltschi.

Joint work with Ron Peled.

Schreiber Building Room 309 at 14:30. Joint work with Ron Peled.

Monday, January 15

Pat Hooper, City College of New York

Refraction in the trihexagonal tiling

**Abstract:**

Refraction in the trihexagonal tiling

I will discuss the dynamics of light rays in the trihexagonal tiling in the plane where triangles and hexagons are transparent and have equal but opposite indices of refraction. Sometimes this is called a `tiling billiards system.' It turns out that almost every light ray is dense in the plane with a periodic family of disjoint open triangles removed. The proof involves some elementary observations about invariant subspaces, an orbit equivalence to straight-line flow on an infinite periodic translation surface, and use of relatively recent results on ergodic theoretic questions for such flows. Most of the talk will be elementary. This talk is based on joint work with Diana Davis and is available at arXiv:1609.00772.

Schreiber Building Room 309 at 14:30. Note special day and time!

Thursday, September 14

Shaked Feldman, Tel Aviv University

Uniform Spanning Trees

**Abstract:**

Uniform Spanning Trees

In this lecture, I will present the two models FUSF and WUSF. I will show some classical results connecting them to electrical currents, random walks and harmonic functions on graphs. After that, I will talk about circle packing, a beautiful combinatorial theorem, that gives rise to many amazing results and methods, from a wide variety of mathematical subjects. With the time left I will talk about recent advancements relating to these models, in particular the results from my soon to be published thesis paper.

Schreiber Building Room 309 at 10:00. Monday, March 13

Mike Hochman, Hebrew University of Jerusalem

How to efficiently code irregular sequences

**Abstract:**

How to efficiently code irregular sequences

Claude Shannon showed that entropy gives the optimal per-symbol coding rate for typical samplkes taken from stationary stochastic processes. Such a sample displays statistical regularity, e.g., the empirical frequencies exist, and describe a sigma-additive probability measure. What happens for sequences without such regularity? In the talk I will answer this in the setting of Borel dynamics and explain why every Borel automorphism without invariant probability measures can be coded with arbitrary efficiency. I would like to focus on some specific examples and explain some of the methods goind into coding of sample paths whose empirical frequencies don't exist.

Schreiber Building Room 309 at 14:30. Monday, March 20

Sébastien Martineau, Weizmann Institute

Locality of critical parameters in statistical mechanics

**Abstract:**

Locality of critical parameters in statistical mechanics

Cayley graphs allow us to get graphs from groups, generalizing the process of getting the hypercubic lattice from Z^{d}: take a group G generated by a finite set S, take the vertex set to be G and put an edge between any g and gs, where g is in G and s is in S. Now, consider a way to assign to each Cayley graph a real number. Can this number be well computed if you are only given the ball of radius 1000 of the Cayley graph under study? This is (essentially) the locality question. I will review the status of this question for the connective constant and the critical parameter of Bernoulli percolation, and explain an application of such results.

This talk will be as self-contained as possible. Includes joint work with Vincent Tassion.

Schreiber Building Room 309 at 14:30. This talk will be as self-contained as possible. Includes joint work with Vincent Tassion.

Monday, March 27

Leonid Pastur, Institute for Low Temperature Physics & Engineering, Ukraine

Analogs of Szegő’s Theorem for Ergodic Operators

**Abstract:**

Analogs of Szegő’s Theorem for Ergodic Operators

We present a setting generalizing that of Szegő’s theorem on the Töplitz operators.
Viewing the strong Szegő’s theorem as an asymptotic trace formula, which is determined by an underlying Töplitz operator and by two functions (the symbol and the
test function), we replace the Töplitz operator by an ergodic operator (e.g. random or
quasiperiodic). In the framework of this setting we consider a variety of asymptotic
formulas different from those known for the Töplitz operators and including certain
central limit theorems and formulas related to quantum information theory.

Schreiber Building Room 309 at 14:30. Monday, April 3

Evgeny Strahov, Hebrew University of Jerusalem

Discrete time determinantal processes related to products of random matrices

**Abstract:**

Discrete time determinantal processes related to products of random matrices

I will talk about a family of random processes in discrete time related to products of random matrices (product matrix processes).
Such processes are formed by singular values of random matrix products, and the number of factors in the random matrix product plays the role of a discrete time.
I will explain that in certain cases product matrix processes are discrete-time determinantal point processes, whose correlation kernels can be expressed in terms of double contour integrals.
In particular, this enables to investigate different scaling limits of the
determinantal process under consideration.

Schreiber Building Room 309 at 14:30. Monday, April 10 and Monday, April 17

Passover break - No seminar.Monday, April 24

Lei Yang, Hebrew University of Jerusalem

Badly approximable points on curves and unipotent orbits in homogeneous spaces

**Abstract:**

Badly approximable points on curves and unipotent orbits in homogeneous spaces

We will study n-dimensional badly approximable points on curves. Given an analytic non-degenerate curve in R^n, we will show that any countable intersection of the sets of weighted badly approximable points on the curve has full Hausdorff dimension. This strengthens a previous result of Beresnevich by removing the condition on weights. Compared with the work of Beresnevich, we study the problem through homogeneous dynamics. It turns out that the problem is closely related to the study of distribution of long pieces of unipotent orbits in homogeneous spaces.

Schreiber Building Room 309 at 14:30. Monday, May 1

Memorial day - No seminar.Monday, May 8

Dominic Yeo, Technion

Frozen percolation with k types

**Abstract:**

Frozen percolation with k types

We study a model for the effect of infections on a population where individuals have one of k types. An inhomogeneous random graph represents the initial connections between these individuals, and over time new connections are made homogeneously, as in the classical random graph process. Each vertex is infected at some rate, resulting in the removal of its entire component. This is a version of a frozen percolation model which (under mild conditions) exhibits self-organised criticality: the dynamics first drive the system to a critical state, and from then on maintain it in criticality. We prove concentration results for the sizes of the components and a local limit theorem, in terms of a multitype branching process whose parameters are critical and described by the solution to an unusual differential equation.

Schreiber Building Room 309 at 14:30. Monday, May 15

Hao Wu, Université de Genève

Hypergeometric SLE and Convergence of Critical Planar Ising Interfaces

**Abstract:**

Hypergeometric SLE and Convergence of Critical Planar Ising Interfaces

Conformal invariance and critical phenomena in two-dimensional statistical physics have been active
areas of research in the last few decades. This talk concerns conformally-invariant random curves that should describe
scaling limits of interfaces in critical lattice models.

The scaling limit of the interface in critical planar lattice models with Doburshin boundary conditions (b.c.), if exists, should satisfy conformal invariance (CI) and domain Markov property (DMP). In 1999, O. Schramm introduced SLE process, and this is the only one-parameter family of random curves with CI and DMP. In 2010, D. Chelkak and S. Smirnov proved that the interface of critical Ising model on the square lattice does converge to SLE(3). In this talk, we discuss the scaling limit of the pair of interfaces in a rectangle with alternating b.c. The scaling limit of the pair of interfaces, if exists, should satisfy CI, DMP and symmetry (SYM). It turns out there is a two-parameter family of random curves satisfying CI, DMP, and SYM, and they are Hypergeometric SLE. For the critical Ising model on the square lattice, the pair of interfaces does converge to Hypergeometric SLE(3). In this talk, we will explain two different proofs for the convergence. Furthermore, we will discuss results about global and local multiple SLEs, which correspond to the scaling limit of the collection of interfaces with alternating b.c. in more general setting.

Schreiber Building Room 309 at 14:30. The scaling limit of the interface in critical planar lattice models with Doburshin boundary conditions (b.c.), if exists, should satisfy conformal invariance (CI) and domain Markov property (DMP). In 1999, O. Schramm introduced SLE process, and this is the only one-parameter family of random curves with CI and DMP. In 2010, D. Chelkak and S. Smirnov proved that the interface of critical Ising model on the square lattice does converge to SLE(3). In this talk, we discuss the scaling limit of the pair of interfaces in a rectangle with alternating b.c. The scaling limit of the pair of interfaces, if exists, should satisfy CI, DMP and symmetry (SYM). It turns out there is a two-parameter family of random curves satisfying CI, DMP, and SYM, and they are Hypergeometric SLE. For the critical Ising model on the square lattice, the pair of interfaces does converge to Hypergeometric SLE(3). In this talk, we will explain two different proofs for the convergence. Furthermore, we will discuss results about global and local multiple SLEs, which correspond to the scaling limit of the collection of interfaces with alternating b.c. in more general setting.

Monday, May 22

Barak Weiss, Tel Aviv University

Random walks on homogeneous spaces and diophantine approximation on fractals

**Abstract:**

Random walks on homogeneous spaces and diophantine approximation on fractals

We extend results of Y. Benoist and J.-F. Quint concerning random walks on homogeneous spaces of simple Lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily Zariski dense, but satisfies some expansion properties for the adjoint action. Using these dynamical results, we study Diophantine properties of typical points on some self-similar fractals in R^d. As examples, we show that for any self-similar fractal K satisfying the open set condition (for instance any
translate or dilate of Cantor’s middle thirds set or of a Koch snowflake), almost every point with respect to the natural measure on K is not badly approximable. Furthermore, almost every point on the fractal is of generic type, which means (in the one-dimensional case) that its continued fraction expansion contains all finite words with the frequencies predicted by the Gauss measure. Joint work with David Simmons.

Schreiber Building Room 309 at 14:30. Monday, May 29

Eliran Subag, Weizmann Institute

The geometry of the Gibbs measure of pure spherical spin glasses

**Abstract:**

The geometry of the Gibbs measure of pure spherical spin glasses

How does a random function on a manifold of very high dimension typically look like? The talk will focus on aspects of this question for the pure spherical spin glass models of statistical mechanics -- namely, random homogeneous polynomials restricted to the N-sphere. First, we will see how the second moment method can be applied to study the distribution of critical points at a given height. Then, we will describe the limiting distribution of the point process associated to critical values (a joint work with Ofer Zeitouni). Finally, we will describe a geometric picture for the Gibbs measure at low temperature: as the dimension tends to infinity, the measure concentrates on "bands" around the critical points of highest values.

Schreiber Building Room 309 at 14:30. Note special day and time!

Sunday, June 4

Noam Berger, Hebrew University of Jerusalem

Harnack inequlity for balanced environments

**Abstract:**

Harnack inequlity for balanced environments

We consider random balanced, not necessarily elliptic, difference equations, and prove a Harnack inequality
in the i.i.d. case. We discuss the relation of this result with random walk and percolation. We then discuss non-i.i.d. cases, and, time permitting, discuss the conjectured continuous analogue of this result.

Based on joint work with M. Cohen, J.-D. Deuschel and X. Guo.

Schreiber Building Room 209, at 12:10. Based on joint work with M. Cohen, J.-D. Deuschel and X. Guo.

Monday, June 12

Matthew Kwan, ETH Zürich

Random designs

**Abstract:**

Random designs

Designs are regular combinatorial structures, generalizing regular graphs and hypergraphs, that have strong connections to a diverse range of different areas of mathematics. Motivated by the flourishing theory of random regular graphs, and some breakthroughs in design theory due to Peter Keevash, the time seems ripe to investigate the subject of random designs. In this talk I'll introduce the topic, and I'll outline a proof of the theorem that almost all "Steiner triple systems" have a "perfect matching".

Schreiber Building Room 309 at 14:30. Monday, June 19

Van Vu, Yale University

Random walks in groups: Local estimates

**Abstract:**

Random walks in groups: Local estimates

Consider the random walk W_k = X_1...X_k where X_i is either M_i or M_i^{-1}, with probability 1/2, where M_1, M_2,.... are fixed invertible complex matrices of a given size. We would like to estimate the probability that W_n = I (or any given matrix, for that matter). Our bound is optimal, up to a constant factor. This can be seen as the non-commutative version of the classical Littlewood-Offord-Erdos bound discovered in the 1940s.

Joint work with T. Pham.

Schreiber Building Room 309 at 14:30. Joint work with T. Pham.

Monday, June 26

Omer Tamuz, Caltech

Large deviations in social learning

**Abstract:**

Large deviations in social learning

Models of information exchange that originate from economics provide interesting questions in probability. We will introduce some of these models, discuss open questions, and explain some recent results.

Joint with Wade Hann-Caruthers and Vadim Martynov.

Schreiber Building Room 309 at 14:30. Joint with Wade Hann-Caruthers and Vadim Martynov.

Monday, October 31

Xiaolin Zeng, Tel Aviv University

Recurrence of two dimensional edge-reinforced random walk

**Abstract:**

Recurrence of two dimensional edge-reinforced random walk

The (linearly) edge-reinforced random walk is a self-interacting random walk in which, at each step, the walker prefers traversing previously visited edges, with a bias proportional to the number of times the edge was traversed. Since its introduction by Diaconis and Coppersmith (1986), the question of understanding the long-time behavior of the walk has received significant attention, and it is natural to expect that the walk is recurrent on recurrent graphs (when simple random walk is recurrent). We prove that the walk is recurrent on Z^2 (for any constant initial weights).
All terms will be explained in the talk. Based on joint works with Sabot and Tarres.

Schreiber Building Room 309 at 14:30. Monday, November 7

Romain Tessera, Université Paris-Sud

Limit shape theorem for first passage percolation on nilpotent groups

**Abstract:**

Limit shape theorem for first passage percolation on nilpotent groups

In a joint work with Itai Benjamini, we prove a limit shape theorem for first passage percolation on a nilpotent group. The idea of proof can be used in the classical setting as well, providing a new (essentially geometric) proof of Alexander's estimates on the speed of convergence to the limit shape.

Schreiber Building Room 309 at 14:30. Monday, November 14

Paul Smith, Tel Aviv University

Towards universality in bootstrap percolation

**Abstract:**

Towards universality in bootstrap percolation

The study of the long-term evolution of monotone cellular
automata on lattices (subject to random initial conditions) dates back
to the late 80s, although it is only recently that a general
definition of such `bootstrap percolation' models has been given and
an ensuing theory begun to be developed. The key question that one
would like to answer is: to what extent do these models, which may
look completely different at the microscopic scale, behave in
`essentially' the same way at the macroscopic scale? In this talk I
shall answer (a precise version of) this question in two dimensions,
and, time permitting, mention some recent work in higher dimensions.

Schreiber Building Room 309 at 14:30. Monday, November 21

Vitali Wachtel, Universität Augsburg

First-passage times for random walks with non-identically distributed increments

**Abstract:**

First-passage times for random walks with non-identically distributed increments

We consider random walks with independent but not necessarily identically distributed increments. Assuming that the increments satisfy the well-known Lindeberg
condition, we investigate the asymptotic behaviour of first-passage times over moving
boundaries. Furthermore, we prove that a properly rescaled random walk conditioned to
stay above the boundary up to time n converges, as n tends to infinity, towards the
Brownian meander.

Schreiber Building Room 309 at 14:30. Monday, November 28

Vincent Delecroix, LaBRI, Bordeaux

Two dynamical generalizations of Hurwitz's theorem in Diophantine approximation

**Abstract:**

Two dynamical generalizations of Hurwitz's theorem in Diophantine approximation

One way of stating Hurwitz's theorem in Diophantine approximation is that among rotations of the interval, the one that achieves the worst recurrence rate is the rotation by the golden mean. We propose two generalizations of this result in a dynamical context. First of all,
Hurwitz's theorem holds in the class of all measurable transformations that
preserve the Lebesgue measure. Secondly, if time permits, we will explain
a generalization of Hurwitz's theorem in the class of interval
exchange transformations.
One important step in proving both theorems is a packing problem
in 2 dimensions.

Schreiber Building Room 309 at 14:30. Monday, December 5

Gordon Slade, University of British Columbia

Critical phenomena and renormalisation group

**Abstract:**

Critical phenomena and renormalisation group

This talk is a continuation of the morning's colloquium, to enter into greater detail about the renormalisation group method.

Schreiber Building Room 309 at 14:30. Monday, December 12

Yair Hartman, Northwestern University

Percolation, Invariant Random Subgroups and Furstenberg Entropy

**Abstract:**

Percolation, Invariant Random Subgroups and Furstenberg Entropy

In this talk I'll present a joint work with Ariel Yadin, in which we solve the Furstenberg Entropy Realization Problem for finitely supported random walks (finite range jumps) on free groups and lamplighter groups. This generalizes a previous result of Bowen. The proof consists of several reductions which have geometric and probabilistic flavors of independent interests.

All notions will be explained in the talk, no prior knowledge of Invariant Random Subgroups or Furstenberg Entropy is assumed.

Schreiber Building Room 309 at 14:30. All notions will be explained in the talk, no prior knowledge of Invariant Random Subgroups or Furstenberg Entropy is assumed.

Monday, December 19

Percy Deift, Courant Institute, New York University

Universality in numerical computations with random data. Analytical results

**Abstract:**

Universality in numerical computations with random data. Analytical results

This is joint work with Tom Trogdon. Here the speaker shows how to prove universality rigorously for the fluctuations in the stopping times for certain, standard numerical algorithms with random data. The proofs rely on recent state of the art results from random matrix theory.

Schreiber Building Room 309 at 14:30. Monday, December 26

Igor Rivin, University of St. Andrews

Statistics of classes of random graphs

**Abstract:**

Statistics of classes of random graphs

We will discuss the distribution of eigenvalues of various classes of random graphs,including geometric graphs, and Cayley and Schreier graphs of finite groups with respect to random generating sets.

Schreiber Building Room 309 at 14:30. Monday, January 2

The Horowitz seminar is held jointly with the Action Now meeting at Tel Aviv University.

The meeting is from 9:30 to 17:00 and features Adrien Boyer, Lionel Levine, Yair Minsky, Yuval Peres and Boris Solomyak. See the schedule here.

The meeting is from 9:30 to 17:00 and features Adrien Boyer, Lionel Levine, Yair Minsky, Yuval Peres and Boris Solomyak. See the schedule here.

Monday, January 9

Daniel El-Baz, Tel Aviv University

Angles in adelic quasicrystals and related problems

**Abstract:**

Angles in adelic quasicrystals and related problems

We discuss some results concerning the local statistics of directions in certain 'adelic quasicrystals'. One such local statistic is the gap distribution, where we build on work by Jens Marklof and Andreas Strömbergsson in the case of (affine) Euclidean lattices and quasicrystals, and by Noam Elkies and Curtis McMullen who studied the fractional parts of sqrt(n). Another is the pair correlation, where we build on joint work with Jens Marklof and Ilya Vinogradov. The methods involve homogeneous dynamics and analytic number theory.

Schreiber Building Room 309 at 14:30. Monday, January 16

Omer Bobrowski, Technion

Topology of Random Geometric Complexes

**Abstract:**

Topology of Random Geometric Complexes

A random geometric complex is an abstract simplicial complex whose vertices are generated by a random point process in a metric space, and higher-order simplexes are added according to some rules that depend on the geometric configuration of the vertices. In this talk we will review recent advances in the study of the homology of random geometric complexes. Loosely speaking, homology is a topological-algebraic structure that contains information about cycles of various dimensions in the complex. We will discuss phase transitions related to “homological connectivity”, as well as the behavior of these complexes in the thermodynamic limit and a higher dimensional notion of percolation.

Schreiber Building Room 309 at 14:30. Monday, January 23

Roland Bauerschmidt, University of Cambridge

Eigenvalue statistics for random regular graphs

**Abstract:**

Eigenvalue statistics for random regular graphs

I will present results on local eigenvalue statistics for random d-regular
graphs. The focus of this talk will be on results on eigenvalue spacing
statistics when d belongs to [N^{o(1)}, N^{2/3-o(1)}]. The talk will be mostly
complementary to my talk at the Workshop on Mathematical Physics on the next day, which will
focus on results in the regime of fixed degree d, to which I will give a brief
outlook (time permitting).

This is joint work with A. Knowles, J. Huang, and H.-T. Yau.

Schreiber Building Room 309 at 14:30. This is joint work with A. Knowles, J. Huang, and H.-T. Yau.

Monday, February 1

Special semester break seminar!

Gaultier Lambert, KTH University, Stockholm

Mesoscopic linear statistics of determinantal processes

**Abstract:**

Mesoscopic linear statistics of determinantal processes

Determinantal point processes arise in the description of eigenvalues of unitary invariant Hermitian random matrices, as well as in many statistical mechanics models such as random tilings, non-intersecting paths, etc. I will explain a cumulant method developed by A. Soshnikov to analyze the asymptotics distributions of linear statistics of determinantal processes. In the mesoscopic regime, within the sine-kernel universality class, we prove that the fluctuations are universal and described by a Gaussian process with H^1/2 correlations. I will also discuss different models which exhibit a transition from Poisson to GUE where non-universal behaviors are observed.

This is a joint work with Kurt Johansson.

Schreiber Building Room 309 at 14:30. This is a joint work with Kurt Johansson.

Monday, February 29

Mikhail Sodin, Tel Aviv University

Spectral measures of {0, 1}-stationary sequences

**Abstract:**

Spectral measures of {0, 1}-stationary sequences

We will discuss what is known and what isn't about
spectral measures of {0, 1}-stationary sequences, and how these
spectral measures are related to some objects of classical
analysis. As usual, we know much less than we would like.

Schreiber Building Room 309 at 14:30. Monday, March 7

Gidi Amir, Bar-Ilan University

Liouville groups with very slowly growing harmonic functions

**Abstract:**

Liouville groups with very slowly growing harmonic functions

A group G has the Liouville property with respect to some generating set S if the only bounded harmonic functions on the Cayley graph of (G,S) are the constant functions. On such Cayley graphs it is interesting to ask how slowly can a non-constant harmonic function grow?
We construct groups with arbitrary slowly growing harmonic functions. More precisely, for any "nice" function f growing slower than log n, we construct a group and a generating set so that there is non-constant harmonic function growing like f, but any harmonic function asymptotically slower than f must be constant.

This is joint work with Gady Kozma.

Schreiber Building Room 309 at 14:30. This is joint work with Gady Kozma.

Monday, March 14

Aser Cortines, Technion

Finite-size corrections to the speed of a branching-selection process

**Abstract:**

Finite-size corrections to the speed of a branching-selection process

We consider a stochastic model of N evolving particles described by a branching
mechanism with selection of the fittest. The model can be seen as the infinite
range limit of a directed polymer in random medium with N sites in the transverse
direction. The particles remain grouped and move like a travelling front driven by
a random noise with a deterministic speed. We focus on the case where the noise
lies in the max-domain of attraction of the Weibull extreme value distribution and
show that under mild conditions the correction to the speed has universal features
depending on the tail probabilities.

Joint work with Francis Comets.

Schreiber Building Room 309 at 14:30. Joint work with Francis Comets.

Monday, March 21

Oren Louidor, Technion

Aging in a logarithmically correlated potential

**Abstract:**

Aging in a logarithmically correlated potential

We consider a continuous time random walk on the box of side length N in Z^2, whose transition rates are governed by the discrete Gaussian free field h on the box with zero boundary conditions, acting as potential: At inverse temperature \beta, when at site x the walk waits an exponential time with mean \exp(\beta h_x) and then jumps to one of its neighbors chosen uniformly at random. This process can be used to model a diffusive particle in a random potential with logarithmic correlations or alternatively as Glauber dynamics for a spin-glass system with logarithmically correlated energy levels. We show that at any sub-critical temperature and at pre-equilibrium time scales, the walk exhibits aging. More precisely, for any \theta > 0 and suitable sequence of times (t_N), the probability that the walk at time t_N(1+\theta) is within O(1) of where it was at time t_N tends to a non-trivial constant as N \to \infty, whose value can be expressed in terms of the distribution function of the generalized arcsine law. This puts this process in the same aging universality class as many other spin-glass models, e.g. the random energy model. Joint work with Aser Cortines-Peixoto and Adela Svejda.

Schreiber Building Room 309 at 14:30. Double seminar! Note special time! Each talk will be 50 minutes.

Monday, March 28

Jon Aaronson, Tel Aviv University

Alexander Glazman, Université de Genève

Aaronson title: Distributional limits of positive, ergodic stationary processes & infinite ergodic transformations

**Aaronson Abstract:**

**Glazman Abstract:**

Alexander Glazman, Université de Genève

Aaronson title: Distributional limits of positive, ergodic stationary processes & infinite ergodic transformations

Every random variable on the positive reals occurs as the
distributional limit of the partial sums some
positive, ergodic stationary process normalized by a
1-regularly varying normalizing sequence (& the process can be chosen
over any EPPT).

I'll try to explain this and (time permitting) some consequences for infinite ergodic theory. Joint work with Benjamin Weiss.

Glazman title: Properties of integrable self-avoiding walksI'll try to explain this and (time permitting) some consequences for infinite ergodic theory. Joint work with Benjamin Weiss.

We consider a self-avoiding walk on the dual Z^2 lattice. This walk can traverse the same square twice but cannot cross the same edge more than once. The weight of each square visited by the walk depends on the way the walk passes through it and the weight of the whole walk is calculated as a product of these weights. We consider a family of critical weights parametrized by angle theta in [pi/3,2pi/3] which satisfy the Yang-Baxter equation. One should view a square with weights corresponding to theta as a rhombus with angle theta.

For theta=pi/3, this can be mapped to the self-avoiding walk on the hexagonal lattice. In this case the connective constant was recently proved to be equal to \sqrt{2+\sqrt{2}}. We show that this proof can be generalized to compute the asymptotic of the partition function for any theta in [pi/3,2pi/3]. Moreover, the 2-point function in the half-plane does not depend on the chosen rhombic tiling as long as all angles are between pi/3 and 2pi/3. This implies that the partition function of bridges of a fixed height tends to 0 as the height tends to infinity. The latter was recently proved for the self-avoiding walk on the hexagonal lattice. In this case we give a new simple proof relying on the parafermionic observable (partly joint work with I. Manolescu).

Schreiber Building Room 309 at 14:10. For theta=pi/3, this can be mapped to the self-avoiding walk on the hexagonal lattice. In this case the connective constant was recently proved to be equal to \sqrt{2+\sqrt{2}}. We show that this proof can be generalized to compute the asymptotic of the partition function for any theta in [pi/3,2pi/3]. Moreover, the 2-point function in the half-plane does not depend on the chosen rhombic tiling as long as all angles are between pi/3 and 2pi/3. This implies that the partition function of bridges of a fixed height tends to 0 as the height tends to infinity. The latter was recently proved for the self-avoiding walk on the hexagonal lattice. In this case we give a new simple proof relying on the parafermionic observable (partly joint work with I. Manolescu).

Monday, April 4

Amos Nevo, Technion

Hyperbolic geometry and pointwise ergodic theorems

**Abstract:**

Hyperbolic geometry and pointwise ergodic theorems

We will describe a completely elementary direct geometric proof of pointwise ergodic theorems for natural averages on the group SL(2,R). We will then describe how this result can be used to generalize the existing pointwise ergodic theorems for averages on isometry groups of hyperbolic spaces beyond the radial case.

Schreiber Building Room 309 at 14:30. Monday, April 11

Erez Nesharim, Tel Aviv University

Existence of Badly Approximable Vectors in Fractals

**Abstract:**

Existence of Badly Approximable Vectors in Fractals

In ergodic dynamical systems almost every point is generic. Many times it is interesting to understand how large is the set of non-generic points. In this talk I will present a criterion for a set to have a nonempty intersection with every “regular” fractal, and mention some applications in Diophantine approximation and dynamics on homogeneous spaces. This talk is based on a joint work with Badziahin, Harrap and Simmons.

Schreiber Building Room 309 at 14:30. Monday, April 18 and Monday, April 25

Replacement day for Wednesday courses and Passover break - No seminar.Monday, May 2

Tal Orenshtein, Université Lyon 1

Router walks on regular trees

**Abstract:**

Router walks on regular trees

Router Walk is a generalization of the Rotor Router model. Initially there is a router configuration on the graph, that is in every vertex there is an infinite sequence of routers which point to its neighbors. Given a router configuration, the walk is deterministic: at each step the walker follows the next unused router in its current location and jumps to a neighbor. In this talk, which is based on a recent joint work with Sebastian Mueller, we shall discuss the problem of recurrence and transience of router walks on regular trees with i.i.d. router sequences.

Schreiber Building Room 309 at 14:30. Monday, May 9

Dmitry Ioffe, Technion

Finite range polymers with on-site repulsion

**Abstract:**

Finite range polymers with on-site repulsion

We discuss polymers on integer lattices, which are modeled
by finite range random walks in repulsive potentials, and which are subject to
pulling forces, confining geometries (e.g. half-spaces or cones ) and interactions
with active substrates.

Based on joint works with Erwin Bolthausen and Yvan Velenik.

Schreiber Building Room 309 at 14:30. Based on joint works with Erwin Bolthausen and Yvan Velenik.

Monday, May 16

Alexander Shamov, Weizmann Institute

Introduction to Gaussian multiplicative chaos

**Abstract:**

Introduction to Gaussian multiplicative chaos

A subcritical Gaussian multiplicative chaos (GMC) is an "exponential" of a generalized (i.e. distributional) Gaussian field, normalized by its expectation. Assuming an appropriate zero-one law, (mixtures of) GMCs are essentially the only integrable random measures that are "local" functionals of the field. In the talk I will review general characterization and convergence results for subcritical GMCs and "logarithmicity"-type necessary conditions for their existence.

Schreiber Building Room 309 at 14:30. Monday, May 23

Alexander Magazinov, Tel Aviv University

Short distance percolation of hard disks

**Abstract:**

Short distance percolation of hard disks

In this talk I will focus on the standard construction of random point arrangements in the plane interacting by a simple hard core exclusion (or, equivalently, on the random packings of unit balls). The intensity parameter λ controls the density of the packing, similarly to that for the Poisson point process.

Given a packing of unit balls, replace each ball by its blow-up to a radius 1 + ε. The union of the blow-ups will be called the excluded volume. A natural question is whether the excluded volume has an infinite component, or, in other words, whether the packing percolates with parameter ε.

I will sketch the proof that a random packing percolates almost surely for every ε > 0 if the intensity λ is large enough, depending only on ε. This settles the problem posed in (Bowen, Lyons, Radin, Winkler, 2006) and extends the result of (Aristoff, 2014), where the case ε > 1/2 was solved.

Schreiber Building Room 309 at 14:30. Given a packing of unit balls, replace each ball by its blow-up to a radius 1 + ε. The union of the blow-ups will be called the excluded volume. A natural question is whether the excluded volume has an infinite component, or, in other words, whether the packing percolates with parameter ε.

I will sketch the proof that a random packing percolates almost surely for every ε > 0 if the intensity λ is large enough, depending only on ε. This settles the problem posed in (Bowen, Lyons, Radin, Winkler, 2006) and extends the result of (Aristoff, 2014), where the case ε > 1/2 was solved.

Double seminar! Note special time! Each talk will be 50 minutes.

Monday, May 30

Felix Pogorzelski, Technion

Naomi Feldheim, Stanford University

Pogorzelski title: Non-commutative quasicrystals

**Pogorzelski Abstract:**

**Feldheim Abstract:**

Naomi Feldheim, Stanford University

Pogorzelski title: Non-commutative quasicrystals

The investigation of aperiodic point sets originates back to work
of Meyer in the 70ies who pursued harmonic analysis on
harmonious sets in Euclidean space. Shechtman's discovery of physical
quasicrystals (1982) via X-ray experiments (diffraction)
triggered a boom of the mathematical analysis of the arising scatter patterns.
In recent work with Michael Björklund and Tobias Hartnick, we developed a
spherical diffraction theory for cut-and-project sets in general lcsc groups,
thus advancing into the non-commutative world.
This seminar aims at describing these quasicrystals, as well as the
dynamical systems which naturally arise from them.
From here, we outline the essentials of a new spherical diffraction theory.

Feldheim title: Mean and Minimum
Let X and Y be two unbounded positive independent random variables.
Write Min_m for the probability of the event {min(X,Y) > m} and Mean_m for that of the event {(X+Y)/2 > m}.
We show that the limit inferior of Min_m / Mean_m is always 0 (as m approaches infinity), regardless of the distributions of X and Y.
We view this statement as a universal anti-concentration result, and discuss several implications.
The proof is elementary but involved, relying on comparison to the "nearest" log-concave measure.
We also provide a multiple-variables, weighted variant of this result in the i.i.d. case and pose a conjectured general result encompassing this phenomenon.

Joint work with Ohad Feldheim.

Schreiber Building Room 309 at 14:10. Joint work with Ohad Feldheim.

Monday, June 6

Brandon Seward, Hebrew University of Jerusalem

The Ornstein isomorphism theorem for countably infinite groups

**Abstract:**

The Ornstein isomorphism theorem for countably infinite groups

The well known Ornstein isomorphism theorem states that if two alphabets have the same Shannon entropy, then the corresponding Bernoulli shifts over the integers are isomorphic. Stepin proved that the class of groups satisfying this theorem is closed under extensions, and Ornstein and Weiss proved that all amenable groups satisfy this theorem. A few years ago Lewis Bowen made significant progress by proving that the isomorphism theorem holds for all countably infinite groups when one assumes that both alphabets have at least 3 letters. In this talk I will show that the isomorphism theorem holds for all countably infinite groups without any additional assumptions.

Schreiber Building Room 309 at 14:30. Monday, October 19

Ryokichi Tanaka, Tohoku University

Random walks on hyperbolic groups: entropy and speed

**Abstract:**

Random walks on hyperbolic groups: entropy and speed

We consider three fundamental quantities which are associated with random walks on groups: entropy, speed and volume growth (exponential growth rate of the group). The fundamental inequality due to Guivarc'h tells that the entropy does not exceed the speed times the volume growth. Vershik (2000) asked about the genuine equality case. We focus on hyperbolic groups, and characterize the equality case;
namely, the equality holds if and only if the harmonic measure and a natural geometric measure--the Hausdorff measure (a Patterson-Sullivan measure)--on the boundary are equivalent. We also discuss open problems related to this question.

I will start with a history of the problem and mention recent progress. All notions will be explained in the talk.

Schreiber Building Room 309 at 14:30. I will start with a history of the problem and mention recent progress. All notions will be explained in the talk.

Monday, October 26

Asaf Nachmias, Tel Aviv University

Indistinguishability of trees in uniform spanning forests

**Abstract:**

Indistinguishability of trees in uniform spanning forests

The uniform spanning forest (USF) of an infinite connected graph G is the weak limit of the uniform spanning tree measure taken on exhausting finite subgraphs of G. It is easy to see that it is supported on spanning graphs of G with no cycles, but it need not be connected. Indeed, a classical result of Pemantle ('91) asserts that when G=Z^d, the USF is almost surely a connected tree if and only if d=1,2,3,4.

We prove that when G is a Cayley graph (or more generally, a unimodular random network) one cannot distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely. This confirms a conjecture of Benjamini, Lyons, Peres and Schramm 2001.

Joint work with Tom Hutchcroft.

Schreiber Building Room 309 at 14:30. We prove that when G is a Cayley graph (or more generally, a unimodular random network) one cannot distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely. This confirms a conjecture of Benjamini, Lyons, Peres and Schramm 2001.

Joint work with Tom Hutchcroft.

Monday, November 2

Nishant Chandgotia, Tel Aviv University

Entropy Minimality and Four-Cycle Free Graphs

**Abstract:**

Entropy Minimality and Four-Cycle Free Graphs

A topological dynamical system (X,T) is said to be entropy minimal if
all closed T-invariant subsets of X have entropy strictly less than
(X,T). In this talk we will discuss the entropy minimality of a
class of topological dynamical systems which appear as the space of
graph homomorphisms from Z^d to graphs without four cycles; for
instance, we will see why the space of 3-colourings of Z^d is entropy
minimal even though it does not have any of the nice topological
mixing properties.

Schreiber Building Room 309 at 14:30. Monday, November 9

Daniel Ueltschi, University of Warwick

The random interchange process on the hypercube

**Abstract:**

The random interchange process on the hypercube

We study random permutations of the vertices of the hypercube. The permutations
are given by products of (uniform, independent) random transpositions on edges.
We establish the existence of a phase transition accompanied by cycles of diverging
lengths. (Joint work with Roman Kotecký and Piotr Miłoś.)

Schreiber Building Room 309 at 14:30. Monday, November 16

Vadim Gorin, MIT

Central Limit Theorem for discrete log-gases

**Abstract:**

Central Limit Theorem for discrete log-gases

A log-gas is an ensemble of N particles on the real line, for which the probability of a configuration is the power of the Vandermonde determinant times the product of a weight w(x) over the positions of particles. Such ensembles are widespread in the random matrix theory, while their discrete counterparts appear in numerous statistical mechanics models such as random tilings and last passage percolation, and also in the asymptotic representation theory. I will explain a new approach which gives Central Limit Theorems for global fluctuations of discrete log-gases for a wide class of the weights w(x). The approach is based on novel discrete equations, which are analogues of the loop equations known in the continuous settings.

Schreiber Building Room 309 at 14:30. Monday, November 23

Yaar Solomon, Stony Brook University

The Danzer problem and a solution to a related problem of Gowers

**Abstract:**

The Danzer problem and a solution to a related problem of Gowers

Is there a point set Y in R^d, and C>0, such that every convex set of volume 1 contains at least one point of Y and at most C? This discrete geometry problem was posed by Gowers in 2000, and it is a special case of an open problem posed by Danzer in 1965. I will present two proofs that answers Gowers' question with a NO. The first approach is dynamical; we introduce a dynamical system and classify its minimal subsystems. This classification in particular yields the negative answer to Gowers' question. The second proof is direct and it has nice applications in combinatorics. The talk will be accessible to a general audience. [This is a joint work with Omri Solan and Barak Weiss].

Schreiber Building Room 309 at 14:30. Monday, November 30

Itai Benjamini, Weizmann Institute

Coarse uniformization and percolation

**Abstract:**

Coarse uniformization and percolation

We will present an elementary problem and a conjecture regarding percolation on planar graphs
suggested by assuming quasi invariance of percolation crossing probabilities under coarse conformal uniformization.

Schreiber Building Room 309 at 14:30. Monday, December 7

Rene Rühr, Tel Aviv University

Distribution of Primitive Points and Shapes of Lattices

**Abstract:**

Distribution of Primitive Points and Shapes of Lattices

Does the set of primitive vectors P(D) on large spheres of radius D in the d-dimensional Euclidean space (d>2) equidistribute when projected on the unit sphere, as D goes to infinity?
A way to address this problem is by making use of homogeneous dynamics: To each set P(D), identify a closed orbit, and use recent measure rigidity results to show that these must equidistribute.
Aka, Einsiedler and Shapira recently considered a refinement of this problem, attaching to each vector the shape of its orthogonal lattice, and showed joint equidistribution in the corresponding ambient space. For dimensions <6 some congruence conditions are assumed. In joint work with Manfred Einsiedler and Philipp Wirth, we give polynomial error rates in D and thereby removing congruence conditions for d=4,5.
The focus of this talk will be on the transition to the dynamical problem, and why quantitative estimates are necessary to give the mere qualitative statement of equidistribution.

Schreiber Building Room 309 at 14:30. Monday, December 14

Mira Shamis, Weizmann Institute

The supersymmetric formalism, applied to the density of states of random matrices

**Abstract:**

The supersymmetric formalism, applied to the density of states of random matrices

I will give an introduction to the supersymmetric formalism.
Then I will show how it can be used to study the eigenvalue
distribution of random matrices. The talk will be self-contained.

Schreiber Building Room 309 at 14:30. Monday, December 21

Ron Rosenthal, ETH Zurich

Eigenvalue confinement and spectral gap for random simplicial complexes

**Abstract:**

Eigenvalue confinement and spectral gap for random simplicial complexes

We consider the adjacency operator of the Linial-Meshulam model for random simplicial complexes on n vertices, where each d-cell is added independently with probability p to the complete
(d-1)-skeleton. From the point of view of random matrix theory, the adjacency matrix is a sparse, self adjoint random matrix with dependent entries. Under the assumption np(1-p) >> log^4 n, we prove
that the spectral gap between the \binom{n-1}{d} smallest eigenvalues and the remaining \binom{n-1}{d-1} eigenvalues is np-2\sqrt{dnp(1-p)}(1+o(1)) with high probability. This estimate follows from a
more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. Based on a joint work with Antti Knowles.

Schreiber Building Room 309 at 14:30. Monday, December 28

Igor Wigman, King's college London

On the number of nodal domains of toral eigenfunctions

**Abstract:**

On the number of nodal domains of toral eigenfunctions

This work is joint with Jerry Buckley.

We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for ``generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions.

Schreiber Building Room 309 at 14:30. We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for ``generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions.

Monday, January 4

Omri Sarig, Weizmann Institute

Measures of maximal entropy for surface diffeomorphisms (joint with Buzzi and Crovisier)

**Abstract:**

Measures of maximal entropy for surface diffeomorphisms (joint with Buzzi and Crovisier)

Newhouse showed that every C infinity diffeomorphism has at least one measure of maximal entropy. We show that in the case of surface diffeomorphisms, the number of such measures is finite, and in the topologically transitive case the measure is unique. This is joint work with Jerome Buzzi and Sylvain Crovisier.

Schreiber Building Room 309 at 14:30. Monday, January 11

Wei Wu, New York University

Quantum triviality of a spin model on uniform spanning forests

**Abstract:**

Quantum triviality of a spin model on uniform spanning forests

Quantum triviality refers to the phenomenon that an interacting
lattice model converges to a free field in the scaling limit. This has
been established for Ising and Phi^4 models, at or above their upper
critical dimension. We describe a simple spin model from uniform spanning
forests in Z^d whose critical dimension is 4 and prove that the scaling
limit is the bi-Laplacian Gaussian field for d≥ 4. At dimension 4,
there is a logarithmic correction for the spin-spin correlation and the
bi-Laplacian Gaussian field is a log correlated field. Based on joint
works with Greg Lawler and Xin Sun.

Schreiber Building Room 309 at 14:30. Note special room and time!

Monday, July 20

Frank den Hollander, Leiden University

Breaking of ensemble equivalence in complex networks

**Abstract:**

Breaking of ensemble equivalence in complex networks

It is generally believed that, for physical systems in the
thermodynamic limit, the microcanonical description in terms of energy coincides with the canonical
description in terms of temperature. However, various examples have been
identified for which the microcanonical and canonical ensembles are not
equivalent. A complete theory of this intriguing phenomenon is still missing.

In this talk we show that ensemble nonequivalence can manifest itself also in discrete enumeration problems. As specific examples, we consider ensembles of graphs with topological constraints. We show that, while graphs with a given number of links are ensemble-equivalent, graphs with a given degree sequence are not. This mathematical result provides a theoretical explanation for various ‘anomalies’ that have recently been observed in real-world networks.

While it is generally believed that ensemble nonequivalence is associated with long-range interactions, our findings show that it may naturally arise in systems with local constraints as well.

Joint work with Diego Garlaschelli, Joey de Mol, Andrea Roccaverde and Tiziano Squartini

Schreiber Building Room 209 at 14:10. In this talk we show that ensemble nonequivalence can manifest itself also in discrete enumeration problems. As specific examples, we consider ensembles of graphs with topological constraints. We show that, while graphs with a given number of links are ensemble-equivalent, graphs with a given degree sequence are not. This mathematical result provides a theoretical explanation for various ‘anomalies’ that have recently been observed in real-world networks.

While it is generally believed that ensemble nonequivalence is associated with long-range interactions, our findings show that it may naturally arise in systems with local constraints as well.

Joint work with Diego Garlaschelli, Joey de Mol, Andrea Roccaverde and Tiziano Squartini

Monday, March 9

Omer Angel, University of British Columbia

Increasing subsequences in random walks

**Abstract:**

Increasing subsequences in random walks

We consider the length of the longest increasing subsequence in an n step simple (and other) random walks, and show that the length is with high probability n^{1/2+o(1)}. Several related problems will be discussed. (with Richard Balka, Yuval Peres).

Schreiber Building Room 309 at 14:30. Monday, March 16

Barak Weiss, Tel Aviv University

Everything is Illuminated (Except for at Most Finitely Many Points)

**Abstract:**

Everything is Illuminated (Except for at Most Finitely Many Points)

Suppose a light source is placed in a polygonal hall of mirrors (so light can bounce off the walls). Does every point in the room get illuminated? This elementary geometrical question was open from the 1950s until Tokarsky (1995) found an example of a polygonal room in which there are two points which do not illuminate each other. Resolving a conjecture of Hubert-Schmoll-Troubetzkoy, in joint work with Lelievre and Monteil we prove that if the angles between walls is rational, every point illuminates all but at most finitely many other points. The proof is based on recent work by Eskin, Mirzakhani and Mohammadi in the ergodic theory of the SL(2,R) action on the moduli space of translation surfaces. The talk will serve as a gentle introduction to the amazing results of Eskin, Mirzakhani and Mohammadi.

Schreiber Building Room 309 at 14:30. Monday, March 23

Tom Hutchcroft, University of British Columbia

Random Hyperbolic Triangulations: Circle Packing and Random Walk

**Abstract:**

Random Hyperbolic Triangulations: Circle Packing and Random Walk

For bounded degree planar graphs, a rich theory has been developed connecting the behaviour of random walk to the geometry of the circle packing embedding. In this talk, I will develop a parallel theory for random triangulations, without the assumption of bounded degree.

First, I will show that the circle packing type (hyperbolic or parabolic) is determined by the average degree of the triangulation, obtaining a new proof of the Benjamin-Schramm Recurrence Theorem.

Secondly, in the hyperbolic case, I will discuss the limiting behaviour of the random walk, showing that the geometric boundary given by the circle packing may be identified with the Poisson boundary of the graph.

Joint work with Omer Angel, Asaf Nachmias and Gourab Ray.

Schreiber Building Room 309 at 14:30. First, I will show that the circle packing type (hyperbolic or parabolic) is determined by the average degree of the triangulation, obtaining a new proof of the Benjamin-Schramm Recurrence Theorem.

Secondly, in the hyperbolic case, I will discuss the limiting behaviour of the random walk, showing that the geometric boundary given by the circle packing may be identified with the Poisson boundary of the graph.

Joint work with Omer Angel, Asaf Nachmias and Gourab Ray.

Monday, March 30 and Monday, April 6

Passover break - No seminar.Monday, April 13

Gidi Amir, Bar-Ilan University

Amenability, entropy and the Liouville property for automaton groups via random walks and electrical networks

**Abstract:**

Amenability, entropy and the Liouville property for automaton groups via random walks and electrical networks

Automaton groups are groups of automorphisms of regular rooted trees that have certain self-similar properties.
These include important examples such as the Grigorchuk group, the basilica group and many others. In this talk we will study the electrical resistance between vertices in some Schreier graphs of these groups and show how these can be used to prove amenability of low activity automaton groups (degree ≤2), as well as deduce the Liouville property and uniform estimates for the entropy of (symmetric, finitely supported) random walks on bounded degree automaton groups.

No prior knowledge of automaton groups, amenability or the Liouville property will be assumed.

This talk is based on joint works with Omer Angel, Nicolas Matte Bon and Balint Virag.

Schreiber Building Room 309 at 14:30. No prior knowledge of automaton groups, amenability or the Liouville property will be assumed.

This talk is based on joint works with Omer Angel, Nicolas Matte Bon and Balint Virag.

Monday, April 20

Benjamin Weiss, The Hebrew University of Jerusalem

Weak mixing properties for non-singular actions

**Abstract:**

Weak mixing properties for non-singular actions

For a probability measure preserving action of a locally compact group G,
there are various characterizations of weak mixing. Analogous definitions
can be given when the action merely preserves the measure class and I will
discuss some implications that hold between these properties (based on joint work with Eli Glasner)

Schreiber Building Room 309 at 14:30. Monday, April 27

Chaim Even Zohar, Hebrew University

Invariants of Random Knots

**Abstract:**

Invariants of Random Knots

Random curves in space and how they are knotted give an insight into the behavior of "typical" knots and links. They have been studied by biologists and physicists in the context of the structure of random polymers. There have been many results obtained via computational experiment, but few explicit computations.

In work with Hass, Linial and Nowik, we study random knots based on petal projections, developed by Adams et al. (2012). We have found explicit formulas for the distribution of the linking number of a two component link. We also find formulas for the moments of two finite type invariants of knots, the Casson invariant and another coefficient of the Jones polynomial.

No background in Knot Theory will be supposed. All terms above will be explained. Joint work with Joel Hass, Nati Linial, and Tahl Nowik.

Schreiber Building Room 309 at 14:30. In work with Hass, Linial and Nowik, we study random knots based on petal projections, developed by Adams et al. (2012). We have found explicit formulas for the distribution of the linking number of a two component link. We also find formulas for the moments of two finite type invariants of knots, the Casson invariant and another coefficient of the Jones polynomial.

No background in Knot Theory will be supposed. All terms above will be explained. Joint work with Joel Hass, Nati Linial, and Tahl Nowik.

Monday, May 4

Mira Shamis, Weizmann Institue

Wegner estimates for deformed Gaussian ensembles, and the Wegner orbital model

**Abstract:**

Wegner estimates for deformed Gaussian ensembles, and the Wegner orbital model

The deformed Gaussian Orthogonal Ensemble is obtained by adding a deterministic
real symmetric matrix to a random matrix drawn from the Gaussian Orthogonal
Ensemble. We shall discuss the behavior of the inverse X to such a random matrix,
and in particular find the sharp magnitude and tails of its norm ||X||, and of the norm
||X u|| for a deterministic vector u.

We discuss applications to the Wegner orbital model, a random operator related to the Anderson model which describes a quantum particle with many internal degrees of freedom (orbitals) in a disordered medium. In particular, we establish localization at strong disorder, with sharp dependence of the localization threshold on the number of orbitals, and prove a uniform bound on the density of states.

Joint work with M. Aizenman, R. Peled, J. Schenker, S. Sodin.

Schreiber Building Room 309 at 14:30. We discuss applications to the Wegner orbital model, a random operator related to the Anderson model which describes a quantum particle with many internal degrees of freedom (orbitals) in a disordered medium. In particular, we establish localization at strong disorder, with sharp dependence of the localization threshold on the number of orbitals, and prove a uniform bound on the density of states.

Joint work with M. Aizenman, R. Peled, J. Schenker, S. Sodin.

Monday, May 11

David Gilat, Tel Aviv University

Convergence in Distribution (D), Convergence in Probability (P) and Almost-Sure convergence (AS) of Martingales. Old Stuff with a New Twist

**Abstract**

Schreiber Building Room 309 at 14:30.

Convergence in Distribution (D), Convergence in Probability (P) and Almost-Sure convergence (AS) of Martingales. Old Stuff with a New Twist

Schreiber Building Room 309 at 14:30.

Note special time!

Monday, May 18

Jonathan Hermon, UC Berkeley

Characterization of cutoff for reversible Markov chains

**Abstract:**

Characterization of cutoff for reversible Markov chains

A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of "worst" (in some sense) sets of stationary measure at least alpha, for some 0 < alpha < 1.

We also give general bounds on the total variation distance of a reversible chain at time t in terms of the probability that some "worst" set of stationary measure at least alpha was not hit by time t. As an application of our techniques we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to infinity.

Joint work with Riddhipratim Basu and Yuval Peres.

Schreiber Building Room 309 at 14:20. We also give general bounds on the total variation distance of a reversible chain at time t in terms of the probability that some "worst" set of stationary measure at least alpha was not hit by time t. As an application of our techniques we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to infinity.

Joint work with Riddhipratim Basu and Yuval Peres.

Monday, May 25

Margherita Disertori, Universität Bonn

Some results on history dependent stochastic processes

**Abstract:**

Some results on history dependent stochastic processes

Edge reinforced random walk (ERRW) and vertex reinforced jump
processes (VRJP) are history dependent stochastic processes,
where the particle tends to come back more often on sites it has
already visited in the past.
For a particular scheme of reinforcement these processes are
random walks in random environment (mixing of reversible Markov chains)
whose mixing measure can be related to a non-linear sigma model introduced
in the context of random matrix models for quantum diffusion.
I will give an overview on these models and explain some recent
results.

Schreiber Building Room 309 at 14:30. Monday, June 1

No seminar due to many participants being away.Audience is recommended to attend Dembo's colloquium talk and Solomyak's analysis seminar talk.

Monday, June 8

Yuval Peres, Microsoft Research

Using random walks to analyze Prediction with Expert Advice

**Abstract:**

Using random walks to analyze Prediction with Expert Advice

We study the classical problem of prediction with expert advice in the adversarial setting with a geometric stopping time. Cover (1965) gave the optimal algorithm that minimizes worst-case regret for the case of 2 experts. In this talk, I will describe the optimal algorithm, adversary and regret for the case of 3 experts. We will see that optimal algorithm for 2 and 3 experts is a probability matching algorithm (analogous to Thompson sampling) against a particular randomized adversary. Remarkably, it turns out that this algorithm is not only optimal against this adversary, but also minimax optimal against all possible adversaries. The analysis of the optimal adversary relies on delicate random walk estimates. At the end of the talk, I will discuss the case of "Bandit feedback", when we just learn the gain of the action we chose, and analyze the effects of imposing a switching cost. This analysis uses a Gaussian Branching random walk.

(Talk based on joint works with Nick Gravin and Balu Sivan and with Ofer Dekel, Jian Ding and Tomer Koren.)

Schreiber Building Room 309 at 14:30. (Talk based on joint works with Nick Gravin and Balu Sivan and with Ofer Dekel, Jian Ding and Tomer Koren.)

Monday, June 15

Ohad Feldheim, University of Minnesota

Long-range order in random three colourings of Z^{d}

**Abstract:**

Long-range order in random three colourings of Z

Consider a random colouring of a bounded domain in Z^{d} with the probability of each colour configuration proportional to exp(-β*N(F)), where β>0 is a parameter (representing the inverse temperature) and N(F) is the number of nearest neighbouring pairs coloured by the same colour. This is the anti-ferromagnetic 3-state Potts model of statistical physics, used to describe magnetic interactions in a spin system. The Kotecký conjecture is that in such a model, for d≥3 and high enough β, a sampled colouring will typically exhibit long-range order, placing the same colour at most of either the even or odd vertices of the box.
We give the first rigorous proof of this fact for large d. This extends previous works of Peled and of Galvin, Kahn, Randall and Sorkin, who treated the case β equals infinity.

The main ingredient in our proof is a new structure theorem for 3-colourings which characterises the ways in which different "phases" may interact, putting special emphasis on the role of edges connecting vertices of the same colour. We also discuss several related conjectures. No background in statistical physics will be assumed and all terms will be explained thoroughly.

Joint work with Yinon Spinka.

Schreiber Building Room 309 at 14:30. The main ingredient in our proof is a new structure theorem for 3-colourings which characterises the ways in which different "phases" may interact, putting special emphasis on the role of edges connecting vertices of the same colour. We also discuss several related conjectures. No background in statistical physics will be assumed and all terms will be explained thoroughly.

Joint work with Yinon Spinka.

Monday, October 27

Uri Shapira, Technion

Escape of mass for measures invariant under the diagonal group

**Abstract:**

Escape of mass for measures invariant under the diagonal group

I will discuss some observations regarding the topology of the space of A-invariant probability measures on the space of lattices (A being the diagonal group in SL_n). In particular, I will explain a construction of Cassels giving rise to a sequence of A-invariant ergodic probability measures (supported on compact A-orbits) which converge to the zero measure (i.e. full escape of mass).

Schreiber Building Room 309 at 14:30. Monday, November 3

Victor Kleptsyn, Institut de Recherche Mathématique de Rennes

Towards rigorous construction for random metrics : the cut-off process

**Abstract:**

Towards rigorous construction for random metrics : the cut-off process

One of the open problems in the domain of quantum gravity is the one of constructing a random metric on a manifold as a limit of a multiplicative cascade; if constructed for the case of a square, it can be thought as the realization of 'exp(DGFF) |dz|’.

Though this problem is well-known, there are very few rigorous known results. One of them is the work of Benjamini and Schramm for the multiplicative cascades on the interval, where the sequence of distances forms a martingale. The (martingale-related) convergence of measures is a key element in a work of Duplantier and Sheffield on the KPZ formula. Finally, the results of Le Gall and Miermont show that one can consider a random metric on the sphere as a limit of random planar maps.

The main result of our work is the rigorous construction of a random metric via multiplicative cascades on hierarchical graphs (like Benjamini's eight graph, dihedral hierarchical lattice, etc.), this situation being both still accessible due to the graph structure, but already complicated due to the high non-uniqueness of candidates for geodesic lines.

A key argument, that allows to find a stationary law for the glueing process, is the cut-off process: instead of looking for a critical renormalization value, we 'stabilize' the process in the supercritical regime by adding a 'shortcut', and then pass to the 'diagonal' limit (the renormalization parameter tends to the critical value, and at the same time the influence of the shortcut tends to zero).

Joint work with M. Khristoforov and M. Triestino.

Schreiber Building Room 309 at 14:30. Though this problem is well-known, there are very few rigorous known results. One of them is the work of Benjamini and Schramm for the multiplicative cascades on the interval, where the sequence of distances forms a martingale. The (martingale-related) convergence of measures is a key element in a work of Duplantier and Sheffield on the KPZ formula. Finally, the results of Le Gall and Miermont show that one can consider a random metric on the sphere as a limit of random planar maps.

The main result of our work is the rigorous construction of a random metric via multiplicative cascades on hierarchical graphs (like Benjamini's eight graph, dihedral hierarchical lattice, etc.), this situation being both still accessible due to the graph structure, but already complicated due to the high non-uniqueness of candidates for geodesic lines.

A key argument, that allows to find a stationary law for the glueing process, is the cut-off process: instead of looking for a critical renormalization value, we 'stabilize' the process in the supercritical regime by adding a 'shortcut', and then pass to the 'diagonal' limit (the renormalization parameter tends to the critical value, and at the same time the influence of the shortcut tends to zero).

Joint work with M. Khristoforov and M. Triestino.

Monday, November 10

Antal Jarai, University of Bath

Sandpiles and the one-end property of uniform spanning forests

**Abstract:**

Sandpiles and the one-end property of uniform spanning forests

The Abelian sandpile model is a stochastic cellular automaton introduced by physicists Bak, Tang and Wiesenfeld (1987) and Dhar (1990). The recurrent states of the model are in bijection with spanning trees of the underlying graph; due to Majumdar and Dhar (1992). This bijection has been very fruitful, as it allows one to translate sandpile problems into questions about uniform spanning trees. When one considers the limit of infinite graphs, an important role is played by the condition that each component of the wired uniform spanning forest has one end; a question studied by Pemantle (1991), Benjamini, Lyons, Peres and Schramm (2001), and Lyons, Morris and Schramm (2008).
I will illustrate the above connection in two recent results: i) the bijection is extended to infinite graphs in a way that preserves the corresponsing probability measures (joint with S. Gamlin); ii) the avalanche exponent is studied for sandpiles on supercritical Galton-Watson trees (joint with W. Ruszel and E. Saada).

Schreiber Building Room 309 at 14:30. Monday, November 17

Gady Kozma, Weizmann Institute

Linearly reinforced random walk

**Abstract:**

Linearly reinforced random walk

We consider a walker which changes its environment as it goes, strengthening each edge it crosses so that it is more likely to cross it again the next time it meets it. The particular case where the reinforcement is linear has a lot of extra structure related to exchangeability and supersymmetry, and enjoys a phase transition in the strength of the reinforcement. We show that when the reinforcement is strong enough the walk enters a condensed phase. Joint work with Angel and Crawford.

Schreiber Building Room 309 at 14:30. Monday, November 24

Omri Sarig, Weizmann Institute

Ergodic properties of the measure of maximal of entropy for smooth three dimensional flows

**Abstract:**

Ergodic properties of the measure of maximal of entropy for smooth three dimensional flows

Geodesic flows on compact connected surfaces of non-positive, non-identically zero curvature are Bernoulli with respect to the Liouville measure (Ornstein-Weiss, Ratner, Pesin). We show that this is the case for the measure of maximal entropy. Some of our results hold for all smooth three dimensional flows.

Joint ongoing work with Ledrappier and Lima.

Schreiber Building Room 309 at 14:30. Joint ongoing work with Ledrappier and Lima.

Monday, December 1

Eitan Bachmat, Ben-Gurion University

A geometric Pollaczek-Khinchine formula involving increasing subsequences, modular transformations and optics

**Abstract:**

A geometric Pollaczek-Khinchine formula involving increasing subsequences, modular transformations and optics

Recently, a couple of airlines experimented with the following boarding policy. Passengers that do not have overhead bin luggage board before passengers who do. In the talk we will analyze this and related policies using the notions presented in the title.

Schreiber Building Room 309 at 14:30. Monday, December 8

Kiran Parkhe, Technion

Groups of polynomial growth and 1D dynamics

**Abstract:**

Groups of polynomial growth and 1D dynamics

Let M be a connected one-manifold, and G a group of homeomorphisms of M which is finitely-generated and virtually nilpotent, i.e., which has polynomial growth. We prove a structure theorem which says, roughly, that the manifold decomposes into wandering regions (in which no G-orbit is dense), and minimal regions (in which every G-orbit is dense); and on the latter, the action is actually Abelian.

As a corollary, if G is a group of polynomial growth of degree d, then for any alpha < 1/d, any continuous G-action on M is conjugate to an action by C^{1 + alpha} diffeomorphisms. This strengthens a result of Farb and Franks.

Schreiber Building Room 309 at 14:30. As a corollary, if G is a group of polynomial growth of degree d, then for any alpha < 1/d, any continuous G-action on M is conjugate to an action by C^{1 + alpha} diffeomorphisms. This strengthens a result of Farb and Franks.

Monday, December 15

Daniel Ueltschi, Warwick

Random loop models and quantum spin systems

**Abstract:**

Random loop models and quantum spin systems

The random loop representations of the quantum Heisenberg models
allow to study these systems using probabilistic methods. They were
introduced twenty years ago by Toth and Aizenman-Nachtergaele.
I will present some rigorous results about a phase transition with long
loops in hypercubes (joint work with R. Kotecky and P. Milos), and
about the decay of certain quantum correlations (joint work with
J. Bjornberg).

Schreiber Building Room 309 at 14:30. Monday, December 22

Michael Krivelevich, Tel Aviv University

The Phase Transition in Site Percolation on Pseudo-Random Graphs

**Abstract:**

The Phase Transition in Site Percolation on Pseudo-Random Graphs

We establish the existence of the phase transition in site percolation on pseudo-random d-regular graphs. Let G=(V,E) be an (n,d,lambda)-graph, that is, a d-regular graph on n vertices in which all eigenvalues of the adjacency matrix, but the first one, are at most lambda in their absolute values. Form a random subset R of V by putting every vertex v in V into R independently with probability p. Then for any small enough constant epsilon>0, if p=(1-epsilon)/d, then with high probability all connected components of the subgraph of G induced by R are of size at most logarithmic in n, while for p=(1+epsilon)/d, if the eigenvalue ratio lambda/d is small enough as a function of epsilon, then typically R contains a connected component of size at least epsilon n/d and a path of length proportional to epsilon^2n/d.

Schreiber Building Room 309 at 14:30. Double seminar! Note special time! Each talk will be 50 minutes.

Monday, December 29

Menny Aka, ETH Zurich

Matan Harel, Université de Genève

Aka title: Diophantine approximation on submanifolds of real matrices and Lie groups

**AKA Abstract:**

**Harel Abstract:**

Matan Harel, Université de Genève

Aka title: Diophantine approximation on submanifolds of real matrices and Lie groups

I will explain how homogeneous dynamics can help to understand Diophantine properties on submanifolds of matrices and on Lie groups. In particular, I'll present notions and results from http://arxiv.org/abs/1307.1489 and http://arxiv.org/abs/1410.3996.

The first answers the question of Diophanticity of nilpotent Lie groups and in particular establishes the first examples of non-Diophantine Lie groups. The second paper aims to compute the critical exponent of Nilpotent Lie groups. To this end we study the Diophantine exponent of analytic submanifolds of mxn real matrices, answering a set of questions of Beresnevich, Kleinbock and Margulis.

Harel title: The Localization Phase Transition in Random Geometric Graphs with Too Many EdgesThe first answers the question of Diophanticity of nilpotent Lie groups and in particular establishes the first examples of non-Diophantine Lie groups. The second paper aims to compute the critical exponent of Nilpotent Lie groups. To this end we study the Diophantine exponent of analytic submanifolds of mxn real matrices, answering a set of questions of Beresnevich, Kleinbock and Margulis.

Consider the Gilbert random geometric graph G(n, r(n)), given by a connecting two points of a Poisson Point Process of intensity n on the unit torus whenever their distance is smaller than the parameter r(n). This model is conditioned on the rare event that the number of edges observed, |E|, is greater than [1 + delta (n)] times its expectation. We show that, when delta is fixed or vanishing sufficiently slowly in n, there exists a "giant clique" with almost all the excess edges forced into the model by the conditioning event. If delta vanishes sufficiently quickly, the largest clique will be, at most, a constant multiple of the usual clique number. Finally, we discuss progress in finding a phase transition function delta_(0)(n), so that when delta is much bigger than delta_(0), the giant clique scenario holds, while delta much smaller than delta_(0) implies no giant clique.

Schreiber Building Room 309 at 14:10. Monday, January 5

Anish Ghosh, Tata Institute of Fundamental Research

Diophantine approximation on varieties

**Abstract:**

Diophantine approximation on varieties

I will discuss the problem of Diophantine approximation on homogeneous varieties of semisimple groups and present analogues of results in classical Diophantine approximation. In particular, I will discuss the problem of counting solutions for Diophantine inequalities on varieties. This is joint work with Alex Gorodnik and Amos Nevo.

Schreiber Building Room 309 at 14:30. Monday, January 12

In-Jee Jeong, Princeton University

Dyadic models in Fluid Dynamics

**Abstract:**

Dyadic models in Fluid Dynamics

Dynamics of fluids are mathematically described by Euler and Navier-Stokes equations. Among many interesting questions regarding these equations, I will describe Kolmogorov's scaling theory and Onsager's conjecture. Then I will describe a toy model for Euler equations, which is an infinite system of simple ODEs. Analogues of Kolmogorov theory and Onsager conjecture will be proved in this toy model.

Schreiber Building Room 309 at 14:30. Monday, January 19

Ofer Zeitouni, Weizmann Institute

Controlled martingales and hitting probabilities

**Abstract:**

Controlled martingales and hitting probabilities

Consider an integer valued martingale S_n=sum_{i=1}^n X_i with bounded step size, with
E(X_i^2 | F_{i-1}) ≥ δ > 0. Is it true that P(|S_n|≤1) ≤ C(δ)/sqrt{n}?

We will explain the (negative) answer to this question and derive sharp upper and lower bounds through a stochastic control formulation.

Based on joint works with Ori Gurel-Gurevich and Yuval Peres, and with Scott Armstrong

Schreiber Building Room 309 at 14:30. We will explain the (negative) answer to this question and derive sharp upper and lower bounds through a stochastic control formulation.

Based on joint works with Ori Gurel-Gurevich and Yuval Peres, and with Scott Armstrong

Monday, July 28

Eviatar B. Procaccia, University of California, Los Angeles

Quenched invariance principle for simple random walk on clusters in correlated percolation models

**Abstract:**

Quenched invariance principle for simple random walk on clusters in correlated percolation models

Quenched invariance principle and heat kernel bounds for random walks on infinite percolation clusters and among i.i.d. random conductances in Z^d were proved during the last two decades.The proofs of these results strongly rely on the i.i.d structure of the models and some stochastic domination with respect to super-critical Bernoulli percolation.
Many important models in probability theory and in statistical mechanics, in particular, models which come from real world phenomena, exhibit long range correlations.
In this talk I will present a new quenched invariance principle, for simple random walk on the unique infinite percolation cluster for a general class of percolation models on Z^d, d≥22, with long-range correlations. This gives new results for random interlacements in dimension d≥3 at every level, as well as for the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime). An essential ingredient of the proof is a new isoperimetric inequality for correlated percolation models.

Joint work with Ron Rosenthal and Artem Sapozhnikov.

Schreiber Building Room 209 at 14:30. Joint work with Ron Rosenthal and Artem Sapozhnikov.

Monday, February 17

Elliot Paquette, Weizmann Institute

Choices and Intervals

**Abstract:**

Choices and Intervals

Consider the following point process on the unit circle. Finitely
many distinct points are placed on the circle in any arbitrary
configuration. This configuration of points subdivides the circle
into a finite number of intervals. At each time step, two points are
sampled uniformly from the circle. Each of these points lands within
some pair of intervals formed by the previous configuration. Add the
point that falls in the larger interval to the existing configuration
of points, discard the other, and then repeat this process.

We will study the behavior of a typical interval, and we will show that as the number of points tends to infinity, this has an almost sure limit, which we characterize. The convergence is established by showing that the size-biased empirical distribution evolves in the limit according to a certain deterministic evolution equation. Although this equation involves a non-local, non-linear operator, it can be studied thanks to a carefully chosen norm with respect to which this operator is contractive. To show the convergence in the presence of noise, we adapt a method of Kushner and Clark to the infinite-dimensional setting.

This is joint work with Pascal Maillard (Weizmann).

Schreiber Building Room 309 at 14:30. We will study the behavior of a typical interval, and we will show that as the number of points tends to infinity, this has an almost sure limit, which we characterize. The convergence is established by showing that the size-biased empirical distribution evolves in the limit according to a certain deterministic evolution equation. Although this equation involves a non-local, non-linear operator, it can be studied thanks to a carefully chosen norm with respect to which this operator is contractive. To show the convergence in the presence of noise, we adapt a method of Kushner and Clark to the infinite-dimensional setting.

This is joint work with Pascal Maillard (Weizmann).

Double seminar! Note special time! Each talk will be 50 minutes.

Monday, February 24

Pascal Maillard, Weizmann Institute

Luca Marchese, Université Paris 13

Maillard title: Contraction of trees

**Maillard Abstract:**

**Marchese Abstract:**

Luca Marchese, Université Paris 13

Maillard title: Contraction of trees

Take a random rooted tree and contract each edge with probability p, where contracting an edge means removing it from the tree and identifying its head and tail. Which random trees are invariant (in law) under this transformation? I will present recent results (joint with Olivier He'nard) which characterize all one-ended trees invariant under this operation, and under the more general operation where edges on the infinite ray are contracted with a possibly different probability q. I will also describe the relationship with (real-valued) self-similar processes and quasi-stationary distributions of linear pure death processes, as well as the common aspects and differences with other renormalization procedures of trees or graphs.

Marchese title: Lagrange spectra for translation surfaces via renormalization
We introduce Lagrange spectra of closed-invariant loci for the action of SL(2,R) on the moduli space of translation surfaces, generalizing the classical Lagrange Spectrum. We treat basic topological issues of spectra, like closure, density of dynamically relevant subsets and existence of an Hall ray. Our approach is renormalization: we get explicit formulas in terms of different continued fraction algorithms. As a consequence, we prove the equivalence of several definitions of bounded Teichmüller geodesics and bounded type interval exchange transformations and moreover we get quantitative estimates for the excursions to the boundary of moduli space in terms of norms of positive matrices in the Rauzy-Veech induction. Joint work with Pascal Hubert and Corinna Ulcigrai.

Schreiber Building Room 309 at 14:10. Monday, March 3

Yinon Spinka, Tel Aviv University

The loop O(n) model

**Abstract:**

The loop O(n) model

A loop configuration on the hexagonal (honeycomb) lattice is a finite subgraph of the lattice in which every vertex has degree 0 or 2, so that every connected component is a cycle. The loop O(n) model on the hexagonal lattice is a probability measure on loop configurations, in which the probability of a configuration is proportional to x^(#edges) n^(#loops) and x,n>0 are parameters called the edge and loop weights. This model arises in statistical physics and is related to the Ising model and self-avoiding walks as well as to the so-called XY and Heisenberg models. We study the loop O(n) model for large values of the loop weight n. We prove that the model undergoes a transition from a disordered to an ordered phase as the edge weight x varies. However, we show that for any value of the edge weight, the probability that the origin is surrounded by a loop of length k decays exponentially in k.

No prior knowledge in statistical mechanics will be assumed and all notions will be explained. Joint work with Hugo Duminil-Copin, Ron Peled and Wojciech Samotij.

Schreiber Building Room 309 at 14:30. No prior knowledge in statistical mechanics will be assumed and all notions will be explained. Joint work with Hugo Duminil-Copin, Ron Peled and Wojciech Samotij.

Monday, March 10

Mike Hochman, The Hebrew University of Jerusalem

Dimension of self-similar sets and additive combinatorics

**Abstract:**

Dimension of self-similar sets and additive combinatorics

I'll present recent progress on the dimension of self-similar sets and measures on the line in the presence of nontrivial "overlaps" in the construction. In particular this resolves a problem of Furstenberg on sums of dilations of certain Cantor sets, and gives information about many other conjectures, e.g. Bernoulli convolutions and the Keane-Smorodinsky {0,1,3} problem. The key new ingredient is to introduce methods from additive combinatorics.

Schreiber Building Room 309 at 14:30. Monday, March 17

Boris Hanin, Northwestern University

Nodal Sets of Random Isotropic Hermite Functions

**Abstract:**

Nodal Sets of Random Isotropic Hermite Functions

Random isotropic Hermite functions of fixed degree in R^d have
an SO(d−1) symmetry and are in some ways analogous to random
spherical harmonics of fixed degree on S^d, whose nodal sets have been the
subject of many recent studies. However, there is a fundamentally new aspect to this ensemble, namely the
existence of allowed and forbidden regions. In the allowed region, the
Hermite functions behave like spherical harmonics, while in the forbidden
region, Hermite functions are exponentially decaying and it is unclear to
what extent they oscillate and have zeros.

The purpose of this talk is to present a new result about the expected volume of the zero set of a random Hermite function in both the allowed and forbidden regions.

This is joint work with Steve Zelditch and Peng Zhou.

Schreiber Building Room 309 at 14:30. The purpose of this talk is to present a new result about the expected volume of the zero set of a random Hermite function in both the allowed and forbidden regions.

This is joint work with Steve Zelditch and Peng Zhou.

Monday, March 24

Idan Perl, Ben-Gurion University

Extinction window of mean field branching annihilating random walk

**Abstract:**

Extinction window of mean field branching annihilating random walk

We study a model of growing population that competes for resources. At
each time step, all existing particles reproduce and the offspring randomly move to
neighboring sites. Then at any site with more than one offspring the particles are
annihilated. This is a non-monotone model, which makes the analysis more difficult.
We consider the extinction window of this model in the finite mean-field case, where
there are n sites but movement is allowed to any site (the complete graph). We
show that although the system survives for exponential time, the extinction window
is logarithmic.

Schreiber Building Room 309 at 14:30. Double seminar! Note special time! Each talk will be 50 minutes.

Monday, March 31

Nick Travers, Technion

Ioan Manolescu, Université de Genève

Travers title: Inversion statistics and longest increasing subsequence for k-card-minimum random permutations

**Travers Abstract:**

**Manolescu Abstract:**

Ioan Manolescu, Université de Genève

Travers title: Inversion statistics and longest increasing subsequence for k-card-minimum random permutations

The k-card-minimum procedure for generating a random permutation of [n] = {1,...,n} is defined as follows. Begin with a deck of n cards labeled 1,...,n and n initially empty positions on a table labeled 1,...,n from left to right. Then, at each time t = 1,...,n choose independently and uniformly at random k cards from the remaining n - t + 1 cards in the deck, remove the lowest numbered (minimum) of these k cards, and place it on the table in position t. The independent sampling is done with replacement, so that the k cards chosen at each time t are not necessarily all distinct.

We prove a weak law of large numbers and central limit theorem for the number of inversions in a random permutation generated according to this procedure, both with fixed k and when k grows sublinearly in n. We also establish the rate of growth of the longest increasing subsequence in a random permutation generated according to this procedure when k grows as a sublinear power of n.

Manolescu title: Planar lattices do not recover from forest fires We prove a weak law of large numbers and central limit theorem for the number of inversions in a random permutation generated according to this procedure, both with fixed k and when k grows sublinearly in n. We also establish the rate of growth of the longest increasing subsequence in a random permutation generated according to this procedure when k grows as a sublinear power of n.

Self-destructive percolation with parameters p, delta is obtained by taking a site percolation configuration with parameter p, closing all sites belonging to the infinite cluster, then opening every site with probability delta, independently of the rest. Call theta(p,delta) the probability that the origin is in an infinite cluster in the configuration thus obtained. For two dimensional lattices, we show the existence of delta > 0 such that, for any p > p_c , theta(p,delta) = 0. This proves a conjecture of van den Berg and Brouwer, who introduced the model. Our results also imply the non-existence of the infinite parameter forest-fire model.

Schreiber Building Room 309 at 14:10. Monday, April 7

Michael Keane, Wesleyan University

The binomial transformation

**Abstract:**

The binomial transformation

In recent years, interest has developed in a special measure preserving transformation, which I call the binomial transformation. In this lecture, I'd like to present two related definitions, and then explain a simple proof that the binomial transformation is ergodic. This proof extends to a larger subclass of adic transformations. It has been conjectured that the binomial transformation is weakly mixing, but no proof of this is known to me.

Schreiber Building Room 309 at 14:30. Monday, April 14

Passover break - No seminar.Monday, April 21

Passover break - No seminar.Monday, April 28

Adela Svejda, Technion

Clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model

**Abstract:**

Clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model

A clock process is the total time that elapses along a given length of the trajectory of a random motion. It is the key object in connection with aging - a phenomenon of random dynamics in random environments whose convergence towards equilibrium becomes increasingly slower. Based on a method due to Durrett and Resnick, Bovier and Gayrard established convergence criteria for clock processes for dynamics on finite graphs. Based on the same method we study clock processes of dynamics in random environments on infinite graphs. As an application we prove that Bouchaud's asymmetric trap model on Z^d, d≥2, ages. This talk is based on joint work with V. Gayrard.

Schreiber Building Room 309 at 14:30. Monday, May 5

Memorial day - No seminar.Monday, May 12

Alexey Gladkich, Tel Aviv University

The cycle structure in random Mallows permutation

**Abstract:**

The cycle structure in random Mallows permutation

The Mallows model is a probability measure on permutations in S_n in which the probability of a permutation pi is proportional to q^{inv(pi)}, where inv(pi) denotes the number of inversions in pi and 0<q<1 is a parameter of the model. The model is an example of a class of distributions called spatial random permutations in which the distribution is biased to be close to the identity in a certain underlying geometry. We study the cycle structure of a permutation sampled from the Mallows model, partially addressing a question of Borodin, Diaconis and Fulman. Our main result is that the expected length of the cycle containing a given point is of order min(1/(1-q)^2, n). In contrast, the expected length of a uniformly chosen cycle is of order min(1/(1-q),n / log(n)). We also overview some related results and conjectures about spatial random permutations. No prior knowledge of random permutations will be assumed.
Joint work with Ron Peled.

Schreiber Building Room 309 at 14:30. Monday, May 19

Ronggang Shi, Tel Aviv University

Pointwise equidistribution for one parameter diagonalizable group action on homogenous space

**Abstract:**

Schreiber Building Room 006 at 14:30.

Pointwise equidistribution for one parameter diagonalizable group action on homogenous space

Let Γ be a lattice in a noncompact simple Lie group L⊂GL_n. Suppose that {g_t} is
a nontrivial one parameter diagonalizable subgroup of L. For certain proper subgroup U of
the unstable horospherical subgroup of {g_t} we show that for any x∈L/Γ the trajectory
{g_t ux: 0≤ t≤ T} is uniformly distributed with respect to the probability Haar measure
of L/Γ as T tends to infinity, for almost every u in U.

Note special room!Schreiber Building Room 006 at 14:30.

Monday, May 26

Zakhar Kabluchko, Universität Ulm

Complex zeros of random partition functions

**Abstract:**

Complex zeros of random partition functions

Let X_1,X_2,... be independent standard normal random variables. The partition function of Derrida's Random Energy Model (REM) at an inverse temperature β is defined as Z_n(β) = ∑_{k=1..2^n} exp(β sqrt(n) X_k).
We study the fluctuations of the random variable Z_n(β) for complex values of β and describe the structure of complex zeros of the random analytic function Z_n as n tends to infinity. The local structure of zeros is described in terms of two interesting objects: the plane Gaussian Analytic Function and a zeta function associated to the Poisson process. We will also study the Generalized Random Energy Model (GREM), a generalization of the Random Energy Model involving long-range correlations between the energies. The phase diagram of the GREM in the complex plane and the distribution of the complex zeros of its partition function will be described. This is joint work with Anton Klimovsky.

Schreiber Building Room 309 at 14:30. Monday, June 2

Jacob Kagan, Weizmann Institute

A translation invariant IDLA model

**Abstract:**

A translation invariant IDLA model

We consider a kind of 2D, directed, tree growing process starting at the line Z. Growth is excluding in the sense that a tree cannot grow further when another tree blocks its way. The question under consideration is whether there exists an infinite tree or not, and we prove the latter.

Ergodicity gives right away that there are either no infinite trees or a positive density of infinite trees on the starting line. The argument then proceeds by coupling the trees with a certain first passage percolation and then deducing that infinite trees do not occur almost surely.

This is a joint work with Noam Berger and Eviatar Procaccia.

Schreiber Building Room 309 at 14:30. Ergodicity gives right away that there are either no infinite trees or a positive density of infinite trees on the starting line. The argument then proceeds by coupling the trees with a certain first passage percolation and then deducing that infinite trees do not occur almost surely.

This is a joint work with Noam Berger and Eviatar Procaccia.

Monday, June 9

Eliran Subag, Weizmann Institute

Extremal point processes and freezing

**Abstract:**

Extremal point processes and freezing

It has been recently shown that the extremal point processes of the branching Brownian motion (BBM), 2-speed BBM, branching random walk, and extremes of the 2D Gaussian free field have the structure of a shifted decorated Poisson point process (SDPPP).
On the other hand, some of those and other related models have been shown to exhibit `freezing' - a phenomenon related to the Laplace functional, which is mostly discussed in physics literature.
We introduce a generalized notion of freezing and show that, under some technical assumptions, it characterizes the structure of SDPPP.

Joint work with Ofer Zeitouni.

Schreiber Building Room 309 at 14:30. Joint work with Ofer Zeitouni.

Monday, October 14

Isaac Meilijson, Tel Aviv University

Placing multiple chips in the roulette table

**Abstract:**

Placing multiple chips in the roulette table

In Dynamic Programming, mixed strategies consist of randomizing the choice of actions. In some problems, such as portfolio management, it makes sense to diversify actions rather than choosing among them purely or randomly. Optimal betting in casinos and roulette by a gambler with fixed goal was studied by Dubins & Savage 1965 and their school without the element of diversification (betting simultaneously on different holes of the roulette), once it was proved (Smith's theorem - Smith 1967, Dubins 1972 and Gilat & Weiss 1976) that diversification doesn't increase the probability of reaching the goal. The scope of this finding is questioned, as it was based on the assumption that the holes on which gamblers can bet are disjoint, such as 1 and BLACK in regular roulette. A counterexample is provided in which holes are nested, such as 1 and RED. Thus, it may be rational for gamblers with fixed goal to place chips on more than one hole at the table.

Schreiber Building Room 309 at 14:30. Monday, October 21

Nick Crawford, Technion

Large Deviation Estimates for Emptiness Formation

**Abstract:**

Large Deviation Estimates for Emptiness Formation

In this talk I will introduce a class of equilibrium quantum systems, called Anisotropic Heisenberg Models, on the d-dimensional integer lattice. In the quantum setting, the Hamiltonian is an operator (rather than a function on phase space). Nevertheless, in 1 dimension in particular and by varying a certain parameter in the definition, it is possible (through path integral expansions) to connect this family of models to a number of more conventional classical particle systems and statistical mechanics models, including the 6 vertex model, self dual 4 state Potts model, Dyson random walk, and the random stirring model.

In finite volume, it is possible to view (the amplitude squared of) the eigenfunctions of the Hamiltonian operator as probability distributions over the collection of all subsets, or configurations of particles, of the fixed volume of interest. The question we study concerns certain "large deviation events" of the ground-state (ie minimal eigenvalue) eigenfunctions. In particular we show, in the thermodynamic limit, that the probability of having no particles in a box of side-length L around the origin decays with L as exp(-cL^{d+1}) (upper and lower bounds). This signifies a very strong correlation structure in the underlying probability measures reminiscent of and connected to large deviations for eigenvalue spacings in GUE.

Joint work with Stephen Ng and Shannon Starr

Schreiber Building Room 309 at 14:30. In finite volume, it is possible to view (the amplitude squared of) the eigenfunctions of the Hamiltonian operator as probability distributions over the collection of all subsets, or configurations of particles, of the fixed volume of interest. The question we study concerns certain "large deviation events" of the ground-state (ie minimal eigenvalue) eigenfunctions. In particular we show, in the thermodynamic limit, that the probability of having no particles in a box of side-length L around the origin decays with L as exp(-cL^{d+1}) (upper and lower bounds). This signifies a very strong correlation structure in the underlying probability measures reminiscent of and connected to large deviations for eigenvalue spacings in GUE.

Joint work with Stephen Ng and Shannon Starr

Monday, October 28

Alexander Gorodnik, University of Bristol

Diophantine approximation by orbits

**Abstract:**

Diophantine approximation by orbits

The classical theory of Diophantine approximation quantifies the
density of rational number in the real line. In a joint work with A.
Ghosh and A. Nevo we consider an analogous problem of approximating by
dense orbits for actions on homogeneous spaces. We explain a general
approach which allows to establish quantitative density and gives the
best possible exponents of approximation in a number of cases.

Schreiber Building Room 309 at 14:30. Monday, November 4

Corinna Ulcigrai, University of Bristol

Ergodic properties of the Ehrenfest model and other infinite periodic billiards

**Abstract:**

Ergodic properties of the Ehrenfest model and other infinite periodic billiards

The Ehrenfest model, introduced in 1912 as a mathematical physics model, is a planar billiard in which a particle moves without friction bouncing elastically at periodically spaced rectangular scatterers. Almost no rigorous results on this model were known until very recently, when several breakthroughs on its recurrence, diffusion and ergodicity properties were achieved, thanks with the connection with the study of periodic translation surfaces.

In this talk, after briefly surveying the known results, we focus on ergodicity. In joint work with K. Fraczek, we show that for almost every direction the Ehrenfest model and other infinite periodic billiards are not ergodic.

Schreiber Building Room 309 at 14:30. In this talk, after briefly surveying the known results, we focus on ergodicity. In joint work with K. Fraczek, we show that for almost every direction the Ehrenfest model and other infinite periodic billiards are not ergodic.

Monday, November 11

Omer Tamuz, Microsoft Research and MIT

Stabilizer Rigidity in Irreducible Group Actions

**Abstract:**

Stabilizer Rigidity in Irreducible Group Actions

An Invariant Random Subgroup (IRS) is a subgroup-valued random variable that is invariant to conjugation. IRSs are probabilistic generalizations of normal subgroups, and share many of their properties. IRSs arise naturally as the stabilizers of measure preserving actions.

In this talk we will show that all irreducible IRSs of product groups, and of higher rank Lie groups, are co-amenable in some normal subgroup. This implies generalizations and strengthenings of similar theorems by Stuck-Zimmer and Bader-Shalom.

Joint work with Yair Hartman.

Schreiber Building Room 309 at 14:30. In this talk we will show that all irreducible IRSs of product groups, and of higher rank Lie groups, are co-amenable in some normal subgroup. This implies generalizations and strengthenings of similar theorems by Stuck-Zimmer and Bader-Shalom.

Joint work with Yair Hartman.

Monday, November 18

Naomi Feldheim, Tel Aviv University

Long gaps between sign-changes of stationary Gaussian processes

**Abstract:**

Long gaps between sign-changes of stationary Gaussian processes

The probability of N independent Gaussians to be all positive is 2^{-N}. In many simple examples, the probability of N stationary Gaussians to be positive is shown to be bounded between exponents in N, thus demonstrating "independent-like" behavior.
Which infinite Gaussian sequences with stationary distribution have this property? How does it relate to their correlations?

Together with Ohad Feldheim, we give a partial answer by presenting broad sufficient conditions for upper and lower exponential bounds on the probability in question. The results hold also for Gaussian stationary functions. Particular cases of this question were addressed by Newell and Rosenblatt in the 1960's and by Antezana, Buckley, Marzo and Olsen in 2012.

Schreiber Building Room 309 at 14:30. Together with Ohad Feldheim, we give a partial answer by presenting broad sufficient conditions for upper and lower exponential bounds on the probability in question. The results hold also for Gaussian stationary functions. Particular cases of this question were addressed by Newell and Rosenblatt in the 1960's and by Antezana, Buckley, Marzo and Olsen in 2012.

Monday, November 25

Roman Kotecký, University of Warwick and Charles University

Long range order for random colourings on planar lattices

**Abstract:**

Long range order for random colourings on planar lattices

We establish a phase transition (entropic long range order) for the uniform random perfect 3-colourings on a class of planar quasi-transitive graphs.
The proof is based on an enhanced Peierls argument (which is of independent interest even for the Ising
model for which it extends the range of temperatures with proven long range order) combined with an additional
percolation argument. The motivation stemming from Potts antiferromagnet models will be explained.

Based on a joint work with Alan Sokal and Jan Swart.

Schreiber Building Room 309 at 14:30. Based on a joint work with Alan Sokal and Jan Swart.

Monday, December 2

Manfred Einsiedler, ETH Zurich

Equidistribution and p-adic dynamics

**Abstract:**

Equidistribution and p-adic dynamics

We will discuss the joint equidistribution of the direction of integer points on large spheres and the shape of the lattice in the orthogonal complement. As we will see this problem can be analyzed using p-adic (or Hecke-) dynamics and theorems of Mozes and Shah resp. Lindenstrauss and myself. We will define all relevant spaces and terms. This is joint work with Aka and Shapira.

Schreiber Building Room 309 at 14:30. Monday, December 9

Jesse Goodman, Technion

The gaps left by a Brownian motion

**Abstract:**

The gaps left by a Brownian motion

Run a Brownian motion in a compact domain for a long time. What
are the shapes of the large gaps remaining? This question, while simple
to state, has surprising connections to harmonic analysis, potential
theory, and complex geometry.

The answer depends strongly on the dimension, and in three or more dimensions the shapes of gaps can be characterised in a largely deterministic way. I will also discuss the conjectured picture in two dimensions, where long range correlations bring the problem into the realm of multiscale, hierarchical models.

Schreiber Building Room 309 at 14:30. The answer depends strongly on the dimension, and in three or more dimensions the shapes of gaps can be characterised in a largely deterministic way. I will also discuss the conjectured picture in two dimensions, where long range correlations bring the problem into the realm of multiscale, hierarchical models.

Monday, December 16

Dima Ioffe, Technion

Low temperature interfaces, layering transitions and Ferrari-Spohn diffusions

**Abstract:**

Low temperature interfaces, layering transitions and Ferrari-Spohn diffusions

2+1 Solid-on-solid interfaces are designed to mimic phase boundaries in genuine three dimensional models of statistical mechanics (e.g. Ising), and to facilitate analysis of various interfacial phenomena, such as fluctuations, entropic repulsion, facet formation and wetting.

For a class of flat SOS type interfaces large scale statistics of level sets, in particular the number of macroscopic size layers, could be quantified in terms of optimal shapes for limiting variational problems. Fluctuations of random level lines around limiting shapes are expected to obey diffusive scaling in the bulk and 1/3-scaling near the edges, which should be reminiscent of 1/3-scaling derived by Ferrari ans Spohn for constrained Brownian motion above circular barriers. Invariance principle to Ferrari-Spohn diffusions is derived for a class of random walks above hard walls with area-type tilts.

Based on joint works with S.Shlosman, F.Toninelli and Y.Velenik.

Schreiber Building Room 309 at 14:30. For a class of flat SOS type interfaces large scale statistics of level sets, in particular the number of macroscopic size layers, could be quantified in terms of optimal shapes for limiting variational problems. Fluctuations of random level lines around limiting shapes are expected to obey diffusive scaling in the bulk and 1/3-scaling near the edges, which should be reminiscent of 1/3-scaling derived by Ferrari ans Spohn for constrained Brownian motion above circular barriers. Invariance principle to Ferrari-Spohn diffusions is derived for a class of random walks above hard walls with area-type tilts.

Based on joint works with S.Shlosman, F.Toninelli and Y.Velenik.

Monday, December 23

Ariel Yadin, Ben Gurion University

Towards an inverse Kleiner Theorem for finitely generated groups

**Abstract:**

Towards an inverse Kleiner Theorem for finitely generated groups

We study Lipschitz harmonic functions on finitely generated groups. (These are naturally probabilistic objects due to the fact that harmonic functions evaluated on a random walk are martingales.) One of the central places where such functions arise is in the "new" proof of Gromov's theorem regarding polynomial growth groups by Kleiner, later simplified by Shalom and Tao. Gromov's theorem asserts that any finitely generated group of polynomial growth is almost nilpotent, thus transforming a seemingly metric property that depends on the choice of generators to an algebraic property. In 2007 Kleiner gave a new proof of this theorem. The main part is to prove that for any finitely generated group of polynomial growth the space of Lipschitz harmonic functions on the group is of finite dimension.

Our point of departure is the study of the possibility of an "inverse" Kleiner's theorem: namely, is it true that if a finitely generated group has a finite dimensional space of Lipschitz harmonic functions, then that group has polynomial growth. This being work in progress, we have not been able to settle this question, but we have made progress towards such a theorem. The main result I will speak about is:

Theorem: If the space of Lipschitz harmonic functions on a finitely generated group G is finite dimensional, then there exists a finite index subgroup H of G such that all Lipschitz harmonic functions restricted to H are homomorphisms of H into the additive complex number group plus a constant (characters + constants).

As a corollary, we obtain that the dimension of the space of Lipschitz harmonic functions on a group is either infinite or some number d that is a group invariant. I will also explain why these are steps towards proving an inverse Kleiner's theorem.

This is joint work with Tom Meyerovitch.

Schreiber Building Room 309 at 14:30. Our point of departure is the study of the possibility of an "inverse" Kleiner's theorem: namely, is it true that if a finitely generated group has a finite dimensional space of Lipschitz harmonic functions, then that group has polynomial growth. This being work in progress, we have not been able to settle this question, but we have made progress towards such a theorem. The main result I will speak about is:

Theorem: If the space of Lipschitz harmonic functions on a finitely generated group G is finite dimensional, then there exists a finite index subgroup H of G such that all Lipschitz harmonic functions restricted to H are homomorphisms of H into the additive complex number group plus a constant (characters + constants).

As a corollary, we obtain that the dimension of the space of Lipschitz harmonic functions on a group is either infinite or some number d that is a group invariant. I will also explain why these are steps towards proving an inverse Kleiner's theorem.

This is joint work with Tom Meyerovitch.

Monday, December 30

Boris Solomyak, University of Washington

On the Fourier asymptotics of self-similar measures

**Abstract:**

On the Fourier asymptotics of self-similar measures

We consider the asymptotics of the Fourier Transform (FT) of Bernoulli convolutions and other self-similar measures. It is known since the work of Erdos and Salem in the 1930-40's that the FT of the Bernoulli convolution with contraction ratio r tends to zero at infinity if and only if 1/r is not a Pisot-Vijayaraghavan (PV) number. However, no quantitative estimates for the decay of the FT follow from the proof. The work of Erdos and Kahane on the power decay for "most" r has recently found new applications, e.g. in the work of P. Shmerkin, who proved absolute continuity of Bernoulli convolutions for
all but zero-dimensional set of parameters r in (0.5.1). I will start with the history, and then discuss some generalizations (joint work with Shmerkin) and applications to the spectral theory of substitution dynamical systems (joint work with A.Bufetov).

Schreiber Building Room 309 at 14:30. Double seminar! Note special time! Each talk will be 50 minutes.

Monday, January 6

Rodrigo Treviño, Tel Aviv University

Keith Merrill, Brandeis University

Treviño title: Controlling deforming geometries

**Treviño Abstract:**

**Merrill Abstract:**

Schreiber Building Room 309 at 14:10. Keith Merrill, Brandeis University

Treviño title: Controlling deforming geometries

Masur’s criterion for unique ergodicity for translation flows on flat surfaces says that if the Teichmuller orbit of a flat surface is recurrent to a compact set of the moduli space, then the translation flow defined by that surface is uniquely ergodic. The recurrence to a compact set in this criterion amounts to the geometry of the surface not degenerating as one applies the Teichmuller deformation.
I will present a theorem which applies to any flat surface of finite area which says that if the geometry of a flat surface undergoing Teichmuller deformation can be more or less controlled, then the translation flow is uniquely ergodic. In the case of the surface being compact, this implies Masur’s criterion and a bit more. The proof is inspired by Forni’s proof for the spectral gap of the Kontsevich–Zorich cocycle. Time permitting, I will discuss the proof (which is in my opinion the more magical part of this) and discuss some possible future directions of this approach.

Merrill title: Diophantine Approximation and Dynamics
In this talk I will discuss the central questions of the field of Diophantine approximation, which seeks to quantify the density of subsets of a metric space. We will focus on the classical case of the density of rational numbers in the real numbers. Then we will discuss and answer these same questions in the context of rational points on spheres and, time permitting, rational points on general quadratic hyper surfaces. The proof techniques involve an analysis of orbits of a certain flow on certain homogeneous spaces. No background knowledge will be assumed.

This is joint work with Lior Fishman, Dmitry Kleinbock, and David Simmons.

This is joint work with Lior Fishman, Dmitry Kleinbock, and David Simmons.

Monday, January 13

Pankaj Vishe, University of York

Inhomogeneous multiplicative Littlewood conjecture and logarithmic savings

**Abstract:**

Inhomogeneous multiplicative Littlewood conjecture and logarithmic savings

We use the dynamics on SL(3,R)/SL(3,Z) to get logarithmic savings in the inhomogeneous multiplicative Littlewood setting. This is a joint work with Alex Gorodnik.

Schreiber Building Room 309 at 14:30. Monday, February 25

Special pre-semester seminar!

Hitoshi Nakada, Keio University, Japan

On cost functions for some Euclidean type algorithms over F_q[X]^d

**Abstract:**

On cost functions for some Euclidean type algorithms over F_q[X]^d

We consider some cost functions of Euclidean type algorithms.
We show that the law of large numbers hold for the bit complexity and the fine
bit complexity of accelerated fully subtractive over F_q -coefficients
d polynomials. For this purpose, we consider the associated dynamical
systems on F_q ((X^{-1})) ^d.

Schreiber Building Room 309 at 14:30. Monday, March 4

Bryna Kra, Northwestern University

Rectangular tiling factors of R^d actions

**Abstract:**

Rectangular tiling factors of R^d actions

We study the space of tilings of R^d for a collection of d-dimensional rectangles. Rudolph showed that there exist 2^d such tiles such that any free measure preserving R^d action has a tiling from this space as a factor map, and we show that d+1 tiles suffices. Furthermore, by studying the geometric properties of the tilings in R^2, we show that this result is sharp in two dimensions. This is joint work with A. Quas and A. Sahin.

Schreiber Building Room 309 at 14:30. Monday, March 11

Cyrille Lucas, Weizmann Institute

Uniform Internal Diffusion Limited Aggregation

**Abstract:**

Uniform Internal Diffusion Limited Aggregation

Internal Diffusion Limited Aggregation, or iDLA, is a growth model in which random sets are constructed recursively. At each step, a random walk starts at the origin and the first point it visits outside the cluster is added to the aggregate. In our modified version of this model, the new random walk starts in a random point distributed uniformly on the aggregate. We will present the major differences between the two models and show the convergence of the normalised Uniform iDLA cluster towards the Euclidean unit ball.

Joint work with Itai Benjamini, Hugo Duminil-Copin, Gady Kozma.

Schreiber Building Room 309 at 14:30. Joint work with Itai Benjamini, Hugo Duminil-Copin, Gady Kozma.

Monday, March 18

Dan Romik, University of California, Davis

Loop percolation, pipe percolation and random noncrossing matchings

**Abstract:**

Loop percolation, pipe percolation and random noncrossing matchings

The talk will be about the same family of random noncrossing matchings introduced in my colloquium talk earlier on the same day, but the discussion will be self-contained and no previous knowledge will be assumed. My goal will be to show that these random noncrossing matchings appear as the connectivity patterns associated with a second (and seemingly unrelated) type of percolation called "pipe percolation". Pipe percolation was previously defined only on a cylindrical geometry (or equivalently as a discrete-time random walk on the finite set of generators of the Temperley-Lieb algebra), but extending the definition to the setting of the entire plane brings up new and subtle issues. I will define the process rigorously as a continuous-time Markov process taking values in the space of noncrossing matchings of the integers. The proof that the construction works will use ideas from ergodic theory and the theory of interacting particle systems. I will also discuss bounds on the matching distance, which relate to the well-known open problem of the "5/48" percolation critical exponent.

Schreiber Building Room 309 at 14:30. Monday, March 25

Passover break - No seminar.Monday, April 1

Passover break - No seminar.Monday, April 8

Ronen Eldan, Weizmann Institute

On the connection between the spectral gap of convex bodies and the variance conjecture.

**Abstract:**

On the connection between the spectral gap of convex bodies and the variance conjecture.

We consider the uniform measure over a high-dimensional isotropic convex body. We prove that, up to logarithmic factors, the isoperimetric minimizers are ellipsoids. Equivalently, we show that up to a logarithmic factor, the "worst-behaving" functions in the corresponding poincare inequality are quadratic functions. We thus establish a connection between two well-known conjectures regarding the uniform measure over a high dimensional convex body, namely the Thin-Shell conjecture and the conjecture by Kannan-Lovasz-Simonovits (KLS) , showing that a positive answer to the former will imply a positive answer to the latter (up to a logarithmic factor). Our proof relies on the analysis of the eigenvalues of a certain random-matrix-valued stochastic process related to a convex body.

Schreiber Building Room 309 at 14:30. Monday, April 15

Memorial day - No seminar.Monday, April 22

Omer Tamuz, Weizmann Institute

The Furstenberg Entropy Realization Problem

**Abstract:**

The Furstenberg Entropy Realization Problem

Random walks on groups and harmonic functions on groups are intimately related to a generalization of measure preserving group actions called "stationary group actions". An important invariant of stationary group actions is their Furstenberg Entropy. The Furstenberg Entropy realization problem is the question of determining the range of possible entropy values realizable for a given random walk.

The talk will include an introduction to this field, an overview of what (little) is known, and some new results. Joint work with Yair Hartman.

Schreiber Building Room 309 at 14:30. The talk will include an introduction to this field, an overview of what (little) is known, and some new results. Joint work with Yair Hartman.

Monday, April 29

Gidi Amir, Bar-Ilan University

Multiple excited random walk, excited Mob and leftover environments

**Abstract:**

Multiple excited random walk, excited Mob and leftover environments

We discuss the model of multiple excited random walk on Z, which is a model of self-interacting random walk on Z which generalizes the notion of random walk in random environment (in this model the walk changes the environment as it walks through it). We will survey results on this model (and higher dimensional analogues) regarding transience, recurrence and positive speed of such walks, and discuss some of the techniques involved. In particular we will discuss some related branching processes and explain how some new observations on these processes can lead to a 0-1 law for directional transience of such walks.

We will then discuss new ideas, that allow us to conclude exact criterions for recurrence, transience and positive speed of such walks on the "leftover" environments - the environments left over after a transient walker goes to infinity. For our analysis we introduce the idea of multiple walkers walking in the same environment, and show how their movement can be analyzed and used to understand the leftover environments.

No prior knowledge on excited random walks or random walk in random environment is assumed.

This is based on joint works with Tal Orenshtein and with Tal Orenshtein and Noam Berger.

Schreiber Building Room 309 at 14:30. We will then discuss new ideas, that allow us to conclude exact criterions for recurrence, transience and positive speed of such walks on the "leftover" environments - the environments left over after a transient walker goes to infinity. For our analysis we introduce the idea of multiple walkers walking in the same environment, and show how their movement can be analyzed and used to understand the leftover environments.

No prior knowledge on excited random walks or random walk in random environment is assumed.

This is based on joint works with Tal Orenshtein and with Tal Orenshtein and Noam Berger.

Monday, May 6

Yuri Lima, Weizmann Institute

A Marstrand theorem for subsets of integers

**Abstract**

Schreiber Building Room 309 at 14:30.

A Marstrand theorem for subsets of integers

Schreiber Building Room 309 at 14:30.

Monday, May 13

Asaf Nachmias, University of British Columbia

Random walks on planar graphs via circle packings

**Abstract:**

Random walks on planar graphs via circle packings

I will describe two results concerning random walks on
planar graphs and the connections with Koebe's circle packing
theorem (which I will not assume any knowledge of):

1. A bounded degree planar triangulation is recurrent if an only if the set of accumulation points of its circle packing is a polar set (that is, has zero logarithmic capacity). This extends a result of He and Schramm who proved recurrence (transience) when the set of accumulation points is empty (a closed Jordan curve). Joint work with Ori Gurel-Gurevich and Juan Souto.

2. The Poisson boundary (the space of bounded harmonic functions) of a transient bounded degree triangulation of the plane is characterized by the topological boundary obtained by circle packing the graph in the unit disk. In other words, any bounded harmonic function on the graph is the harmonic extension of some measurable function on the boundary of the unit disc. Joint work with Omer Angel, Martin Barlow and Ori Gurel-Gurevich.

Schreiber Building Room 309 at 14:30. 1. A bounded degree planar triangulation is recurrent if an only if the set of accumulation points of its circle packing is a polar set (that is, has zero logarithmic capacity). This extends a result of He and Schramm who proved recurrence (transience) when the set of accumulation points is empty (a closed Jordan curve). Joint work with Ori Gurel-Gurevich and Juan Souto.

2. The Poisson boundary (the space of bounded harmonic functions) of a transient bounded degree triangulation of the plane is characterized by the topological boundary obtained by circle packing the graph in the unit disk. In other words, any bounded harmonic function on the graph is the harmonic extension of some measurable function on the boundary of the unit disc. Joint work with Omer Angel, Martin Barlow and Ori Gurel-Gurevich.

Monday, May 20

Omer Angel, University of British Columbia

Evolving sets and hitting times

**Abstract:**

Evolving sets and hitting times

I will review the method of evolving sets, describe a new connection to hitting times, and show that in certain cases, permutations can only accelerate hitting.

Schreiber Building Room 309 at 14:30. Monday, May 27

Pascal Maillard, Weizmann Institute

Performance of the Metropolis algorithm on a disordered tree

**Abstract:**

Performance of the Metropolis algorithm on a disordered tree

I will present recent results obtained with Ofer Zeitouni on the performance of the Metropolis algorithm on a branching random walk tree. Based on our recent article available at arXiv:1304.0552.

Schreiber Building Room 309 at 14:30. Monday, June 3

John Smillie, Cornell University

Do analogs of Ratner's theorems hold for strata of translation surfaces?

**Abstract:**

Do analogs of Ratner's theorems hold for strata of translation surfaces?

Questions about polygonal billiards lead to the study of moduli spaces of translation surfaces or "strata". There are intriguing parallels between strata and homogeneous spaces and there has been a fruitful migration of ideas from homogeneous dynamics to "Teichmuller dynamics" on strata. One pivotal question is the extent to which Ratner theory applies to strata. I will discuss positive results of Eskin and Mirzakhani and describe negative results obtained with Barak Weiss.

Schreiber Building Room 309 at 14:30. Monday, June 10

Amir Dembo, Stanford University

Persistence Probabilities

**Abstract:**

Persistence Probabilities

Persistence probabilities concern how likely it is that a stochastic process has a long excursion above a fixed level and what are the relevant scenarios for this behavior. Power law decay is expected in many cases of physical significance and the issue is to determine its power exponent parameter. I will survey recent progress in this direction (jointly with Jian Ding, Fuchang Gao, and Sumit Mukherjee),
dealing with random algebraic polynomials of independent coefficients, iterated partial sums and other auto-regressive sequences, and with the solution to heat equation initiated by white noise.

Schreiber Building Room 309 at 14:30. Monday, June 17

Elchanan Mossel, University of California, Berkeley

Robust Optimality of Gaussian Noise Stability

**Abstract:**

Robust Optimality of Gaussian Noise Stability

In 1985 C. Borell proved that under the Gaussian measure, half-spaces are the most stable sets.
I will present two new proofs of this result. The first proof solves a long standing open problem by showing
that half-spaces are the unique optimizers. It also provides
quantitative dimension independent versions of uniqueness, showing that a set which is almost optimally noise stable must be close to a half-space. This has a number of application in theoretical computer science and social choice. The first proof also allows to answer a long standing open problem by Ledoux. The second proof is proved by induction on dimension in the discrete cube which allows to derive a proof of "Majority is Stablest" in the "Sum of Squares Proof System" thus aswering a recent question regarding semi-definite relaxations. Based on joint works with J. Neeman and Anindya De.

Schreiber Building Room 309 at 14:30. Monday, June 24

Pascal Vanier, Hebrew University

Characterizations of periods of multidimensional SFTs

**Abstract:**

Characterizations of periods of multidimensional SFTs

In this talk, we will introduce several notions of periodicity for higher dimensional shifts and give computational characterizations of sets of periods of SFTs for some of them. We will in particular focus on points with a finite orbit, the straightforward generalization of periodicity from dimension 1. We will show that sets of periods of multidimensional SFTs correspond exactly to the languages of the computational complexity class NE (languages recognizable in nondeterministic linear exponential time). In order to attain this result we will first focus on the intermediate notion of horizontal periodicity, whose characterization is easier to achieve.

Schreiber Building Room 309 at 14:30. Monday, October 15

Special pre-semester seminar!

Ram Band, University of Bristol

Spectral Geometry on Graphs

**Abstract:**

Spectral Geometry on Graphs

The talk's theme is the extraction of geometric information about graphs
(metric or combinatorial) from the spectra of the graph's Schroedinger
operators (continuous or discrete), and from the distribution of sign
changes on the corresponding eigenfunctions.
These include questions such as e.g., the ability to "hear the shape of
the graph";
the extent to which the spectral sequence and the sequence of the number
of sign changes (or number of nodal domains) complement or overlap each
other;
the derivation of topological information from the study of the response
of the spectrum to variation of scalar or magnetic potentials on the
graph, etc.

In the present talk I shall illustrate this research effort by reviewing several results I obtained recently. The first example answers the question "Can one count a tree?" which appears in the following context: It is known that the number of sign changes of the eigenfunction on tree graphs equals to the position of the corresponding eigenvalue in the spectrum minus one. Is the reverse true? If yes, one can tell a tree just by counting the number of its sign changes. For the proof I shall introduce an auxiliary magnetic field and use a very recent result of Berkolaiko and Colin de Verdiere to connect the spectrum and the number of sign changes. Next, I will discuss the band spectrum obtained by varying the magnetic phases on the graph. I will prove that the magnetic band-to-gap ratio (quality of conductance) is a universal topological quantity of a graph. This result highlights the spectral geometric importance of this invariant and sheds a new light on previous works about periodic potentials on graphs.

The talk contains content of a work in progress with Gregory Berkolaiko. All concepts will be explained and no previous knowledge of the topic is required.

Schreiber Building Room 309 at 14:30. In the present talk I shall illustrate this research effort by reviewing several results I obtained recently. The first example answers the question "Can one count a tree?" which appears in the following context: It is known that the number of sign changes of the eigenfunction on tree graphs equals to the position of the corresponding eigenvalue in the spectrum minus one. Is the reverse true? If yes, one can tell a tree just by counting the number of its sign changes. For the proof I shall introduce an auxiliary magnetic field and use a very recent result of Berkolaiko and Colin de Verdiere to connect the spectrum and the number of sign changes. Next, I will discuss the band spectrum obtained by varying the magnetic phases on the graph. I will prove that the magnetic band-to-gap ratio (quality of conductance) is a universal topological quantity of a graph. This result highlights the spectral geometric importance of this invariant and sheds a new light on previous works about periodic potentials on graphs.

The talk contains content of a work in progress with Gregory Berkolaiko. All concepts will be explained and no previous knowledge of the topic is required.

Monday, October 22

Yinon Spinka, Tel Aviv University

The Random Graph-Homomorphism Model

**Abstract:**

The Random Graph-Homomorphism Model

We discuss the random graph-homomorphism model introduced by Benjamini, Häggström and Mossel (2000). This model generalizes the simple random walk on Z, and therefore, is sometimes referred to as a G-indexed random walk or a random height function on a graph.
We begin by introducing the model (no prior knowledge is assumed) and by giving some examples. We then give a short survey of known results and discuss some conjectures. The main questions of interest are the variance at a given vertex and the range a of random homomorphism.
We then continue on to our results. We study two certain families of graphs (which can be seen as modifications of a torus or a line segment) and show a sharp transition in the behavior of a typical homomorphism. The main results will be stated and outlines and ideas of some proofs will be given. This work answers a question posed by Benjamini, Yadin and Yehudayoff (2007).

Schreiber Building Room 309 at 14:30. Monday, October 29

Michael Krivelevich, Tel Aviv University

Random subgraphs of large minimum degree graphs

**Abstract:**

Schreiber Building Room 309 from 14:30 to 16:00!

Random subgraphs of large minimum degree graphs

Consider the following very general model of random graphs: let G be a finite graph of minimum degree at least k, for k tending to infinity, and form a random subgraph G_p of G by taking each edge of G with probability p=p(k), independently. What can be said about typical properties of such random graph? This model covers a lot of ground, including binomial random graphs G(k+1,p), random subgraphs of the k-dimensional binary cube Q^k, random subgraphs of k-regular expanders etc. Generality has its price, and some clasical questions from the theory of random graphs (appearance of a fixed subgraph, chromatic number etc.) become irrelevant, while some others are probably just too hard. Still, there is quite a number of attractive problems that appear to be approachable. In this talk, I will report about our recent results on some of them.

Based on joint works with (subsets of) Alan Frieze, Choongbum Lee, Benny Sudakov.

Longer seminar! Note special time!Based on joint works with (subsets of) Alan Frieze, Choongbum Lee, Benny Sudakov.

Schreiber Building Room 309 from 14:30 to 16:00!

Monday, November 5

Itai Benjamini, Weizmann Institute

Euclidean vs graph metric

**Abstract:**

Euclidean vs graph metric

We will discuss how well large (random and not random) graphs approximate other spaces such as the Euclidean plane.

Schreiber Building Room 309 at 14:30. Monday, November 12

Yogeshwaran Dhandapani, Technion

On the topology of some random complexes built over stationary point processes

**Abstract:**

On the topology of some random complexes built over stationary point processes

There has been recent interest in understanding the homology of random simplicial complexes built over point processes, primarily motivated by problems in applied algebraic topology. I shall describe our new results about the growth of homology groups of Cech and Vietoris-Rips complexes built over general stationary point processes. Both these complexes have points of the point process as vertices and the faces are determined by some deterministic geometric rule. The aim of the talk shall be to explain the quantitative differences in the growth of homology groups measured via Betti numbers between the Poisson point process and other point processes which exhibit repulsion such as the Ginibre ensemble, zeros of Gaussian analytic functions, perturbed lattice etc. I shall also try to hint at the proof techniques which involve detailed analysis of subgraph and component counts of the associated random geometric graphs and are applicable to similar functionals of point processes such as Morse critical points. This is a joint work with Prof. Robert Adler.

Schreiber Building Room 309 at 14:30. Monday, November 19

Benjamin Weiss, The Hebrew University of Jerusalem

On "a curious conjugacy invariant" of Halmos

**Abstract:**

On "a curious conjugacy invariant" of Halmos

One of the problems that Halmos posed at the end
of his classic lectures on ergodic theory concerns
a curious conjugacy invariant that he suggested
might distinguish between ergodic toral automorphisms.
I will describe this invariant and report on some
recent results of A. Quas and T. Soo on the ergodic
universality of toral automorphisms which relate to this
invariant. In addition I will indicate an alternative argument
which will suffice for the application to the Halmos
problem.

Schreiber Building Room 309 at 14:30. Monday, November 26

Ron Rosenthal, The Hebrew University of Jerusalem

Random walks, spectrum and homology of complexes

**Abstract:**

Random walks, spectrum and homology of complexes

There are well known connections between the random walk on a graph, and its topological and
spectral properties. Here we define a stochastic process on higher dimensional simplicial complexes,
which reflects their homological and spectral properties in a parallel way. This leads to high
dimensional analogues (not all of which hold!) of classical theorems of Kesten, Alon-Boppana, and
others. No previous knowledge is assumed. Joint work with Ori Parzanchevski.

Schreiber Building Room 309 at 14:30. Monday, December 3

Action Now meeting at Bar-Ilan University - No seminar.Monday, December 10

Brad Rodgers, University of California, Los Angeles

Statistics of the zeros of the zeta function: mesoscopic and macroscopic phenomena

**Abstract:**

Statistics of the zeros of the zeta function: mesoscopic and macroscopic phenomena

We review the well known microscopic correspondence between random zeros of the Riemann zeta-function and the eigenvalues of a random unitary or Hermitian matrix, and discuss evidence that this correspondence extends to larger mesoscopic collections of zeros or eigenvalues. In addition, we discuss interesting phenomena that appears in the statistics of even larger macroscopic collections of zeros. The terms microscopic, mesoscopic, and macroscopic are from random matrix theory and will be defined in the talk. If time permits we will raise some questions about general classes of point processes, which model the distribution of zeros.

This talk is based in part on results in the papers arXiv: 1203.3275 math.NT, and arXiv: 1205.0303 math.PR.

Schreiber Building Room 309 at 14:30. This talk is based in part on results in the papers arXiv: 1203.3275 math.NT, and arXiv: 1205.0303 math.PR.

Monday, December 17

Oren Louidor, University of California, Los Angeles

Isoperimetry in Supercritical Percolation.

**Abstract:**

Isoperimetry in Supercritical Percolation.

We consider the unique infinite connected component of supercritical bond percolation on the square lattice and study the geometric properties of isoperimetric sets, i.e., sets with minimal boundary for a given volume. For almost every realization of the infinite connected component we prove that, as the volume of the isoperimetric set tends to infinity, its asymptotic shape can be characterized by an isoperimetric problem in the plane with respect to a particular norm. As an application we then show that the anchored isoperimetric profile with respect to a given point as well as the Cheeger constant of the giant component in finite boxes scale to deterministic quantities. This settles a conjecture of Itai Benjamini for the plane. Joint work with M. Biskup, E. Procaccia and R. Rosenthal.

Schreiber Building Room 309 at 14:30. Monday, December 24

Yuval Peres, Microsoft Research

Balanced self-interacting random walks.

**Abstract:**

Balanced self-interacting random walks.

It is well known that a random walk in d>2 dimensions where the steps are i.i.d. mean zero and fully supported (not restricted to a hyperplane), is transient.
In a recent elegant paper, Benjamini, Kozma and Schapira (2011) asked if we still must have transience when each step is chosen from either \mu_1 or \mu_2 based on the past, where \mu_1 and \mu_2 are fully supported mean zero distributions in dimension d>2. (e.g. we could use \mu_1 if the current state has been visited before, and \mu_2 otherwise), We answer their question, and show the answer can change when we have three measures instead of two. To prove this, we will adapt the classical techniques of Lyapunov functions and excessive measures to this setting. No prior familiarity with these methods will be assumed, and they will be explained in the talk. Many open problems remain in this area, even in two dimensions.

Lecture based on joint work with Serguei Popov (Campinas, Brazil) and Perla Sousi (Cambridge, UK).

Schreiber Building Room 309 at 14:30. Lecture based on joint work with Serguei Popov (Campinas, Brazil) and Perla Sousi (Cambridge, UK).

Monday, December 31

Fedor Nazarov, Kent State University

Littlewood-Offord-Turan estimate for the number of real zeroes of a random polynomial with i.i.d. coefficients

**Abstract:**

Schreiber Building Room 309 from 14:30 to 16:00!

Littlewood-Offord-Turan estimate for the number of real zeroes of a random polynomial with i.i.d. coefficients

We show that the average number of real zeroes of
any random polynomial of degree $n$ with independent identically
distributed coefficients does not exceed $C\log^4 n$
with some absolute $C>0$. The proof follows closely
the paper by Littlewood and Offord published in 1942,
where the case of $\pm 1$ coefficients was considered.
Our main deviation from their scheme is using the Turan lemma
from Turan's 1953 book in place of the pointwise
"anticoncentration" estimate used by Littlewood and Offord.
The result is on par with the original Littlewood-Offord
bound $C\log^2 n$ but still short of the would be
optimal estimate $C\log n$, which is widely
believed to hold and whose validity is known for all
sufficiently regular distributions. This is a joint work with Mikhail Sodin.

Longer seminar! Note special time!Schreiber Building Room 309 from 14:30 to 16:00!

Monday, January 7

Tom Meyerovitch, Ben-Gurion University

On equivariant thinning, allocation and matching schemes for Poisson processes

**Abstract:**

On equivariant thinning, allocation and matching schemes for Poisson processes

Poisson thinning, allocation and matching are ``natural'' operations on a realization of a Poisson process which have been considered in the literature. A Poisson thinning, for instance, is a rule for selecting a subset of the points in the Poisson process which are equal in distribution to a lower intensity Poisson process. There are interesting and non-trivial constructions of isometry-equivariant Poisson thinning, Poisson allocation and Poisson matching.

In this talk I will consider the existence of operations of the above types which are equivariant with respect to a group of measure-preserving transformations other than isometries. Evans proved that the only linear transformations which admit equivariant Poisson thinning are isometries. I will show that no equivariant thinning, allocation or matching is possible with respect to any conservative and ergodic measure preserving transformation. My proof is based on an ergodicity result which uses Kean's classic ``ergodic multiplier theorem''. The definitions involved will be given during the talk, no background with Poisson processes is assumed.

Schreiber Building Room 309 at 14:30. In this talk I will consider the existence of operations of the above types which are equivariant with respect to a group of measure-preserving transformations other than isometries. Evans proved that the only linear transformations which admit equivariant Poisson thinning are isometries. I will show that no equivariant thinning, allocation or matching is possible with respect to any conservative and ergodic measure preserving transformation. My proof is based on an ergodicity result which uses Kean's classic ``ergodic multiplier theorem''. The definitions involved will be given during the talk, no background with Poisson processes is assumed.

Monday, January 14

Jay Rosen, The City University of New York

Markovian loop soups, permanental processes and isomorphism theorems

**Abstract:**

Markovian loop soups, permanental processes and isomorphism theorems

We show how to construct loop soups for general Markov processes and explain how loop soups offer a deep understanding of
Dynkin's isomorphism theorem, and beyond.

Schreiber Building Room 309 at 14:30. Monday, January 21

Piotr Miłoś, University of Bath and University of Warsaw

Heavy-tailed branching Lévy motion in inhomogeneous potential

**Abstract:**

Heavy-tailed branching Lévy motion in inhomogeneous potential

In my talk I will present results concerning a branching
system as follows. Particles evolve according to a real-valued Lévy
motion with heavy tails and branch according to a dyadic branching
law. We consider three different regimes of the branching rate:

- constant in space, then the system size grows exponentially.

- \beta \log(x), then the system size grows as exp(exp(\beta/(e \alpha)t)).

- \log^{1+\epsilon}(x), then almost surely the system explodes, i.e. produces an infinite number of particles in finite time.

These are joint results with Simon Harris (University of Bath)

Schreiber Building Room 309 at 14:30. - constant in space, then the system size grows exponentially.

- \beta \log(x), then the system size grows as exp(exp(\beta/(e \alpha)t)).

- \log^{1+\epsilon}(x), then almost surely the system explodes, i.e. produces an infinite number of particles in finite time.

These are joint results with Simon Harris (University of Bath)

Monday, July 2

Oren Louidor, University of California, Los Angeles

Complete Survival for the Biased Voter Model on Regular Trees

**Abstract:**

Complete Survival for the Biased Voter Model on Regular Trees

We consider the Biased Voter Model (also known as the Williams Bjerknes model)
on the d-regular tree and prove that local survival implies asymptotic consensus (or complete survival). Complete survival and fixation are also considered in the case of general vertex transitive graphs, where the initial configuration is chosen according to a distribution which is automorphism invariant and/or ergodic. Joint work with Ran Tessler and Alexander Vandenberg-Rodes.

Schreiber Building Room 309 at 14:30. Monday, July 9

Brendan Farrell, California Institute of Technology

The Jacobi ensemble and discrete uncertainty principles

**Abstract:**

The Jacobi ensemble and discrete uncertainty principles

The Jacobi ensemble is the third classical ensemble of random
matrix theory: it describes the eigenvalue distribution of a form of
Gaussian random matrix known as a MANOVA matrix. Matrices of this form
are related to angles between random subspaces and, in particular, to
discrete uncertainty principles. We present initial universality results for
MANOVA matrices and for random submatrices of the discrete Fourier transform
matrix. This is partially joint work with László Erdős.

Schreiber Building Room 008 at 14:30. Monday, August 20

Todd Kemp, University of California, San Diego

Liberation of Random Projections

**Abstract**

Note special time!

Schreiber Building Room 309 at 13:00.

Liberation of Random Projections

Note special time!

Schreiber Building Room 309 at 13:00.

Monday, March 5

Barak Weiss, Ben-Gurion University

The Mordell-Gruber spectrum and homogeneous dynamics.

**Abstract:**

The Mordell-Gruber spectrum and homogeneous dynamics.

Given a lattice in euclidean space, its Mordell constant is the supremum
of the volumes (suitably normalized) of central symmetric boxes with sides
parallel to the axes, containing no nonzero lattice points. This is a
well-studied quantity in the geometry of numbers, and is invariant under
the action of diagonal matrices on the space of lattices. In joint work
with Uri Shapira, using dynamics of the diagonal group we obtain new
results about the set of possible values of the Mordell constant (the
so-called Mordell-Gruber spectrum).

Schreiber Building Room 309 at 14:30. Monday, March 12

Noam Berger, The Hebrew University of Jerusalem

New conditions for ballisticity for RWRE

**Abstract:**

New conditions for ballisticity for RWRE

In 2000, Sznitman proved that a uniformly elliptic RWRE is ballistic if a certain back-stepping probability decays
exponentially. Later he weakened this condition to stretched exponential decay. In the talk I will show that in dimensions
four and higher, the condition can be relaxed to (high degree) polynomial decay. All terminology will be explained in the talk.
Based on joint work with Drewitz (Zurich) and Ramirez (Santiago de Chile).

Schreiber Building Room 309 at 14:30. Monday, March 19

Agelos Georgakopoulos, Technical University, Graz

Random walks, electrical networks, and the role of the Poisson boundary

**Abstract:**

Random walks, electrical networks, and the role of the Poisson boundary

I will present some basic facts concerning electrical networks and their connections to random walk. Then, I will present a new result relating Dirichlet harmonic functions on an infinite graph to its Poisson boundary. The talk will be accessible to the non-expert.

Joint work with V. Kaimanovich.

Schreiber Building Room 309 at 14:30. Joint work with V. Kaimanovich.

Monday, March 26

Jon Aaronson, Tel Aviv University

On the categories of weak mixing in infinite measure spaces

**Abstract:**

On the categories of weak mixing in infinite measure spaces

I will discuss some propositions such as "in general" an infinite measure preserving trasformation is subsequence rationally weakly mixing, but not rationally weakly mixing.

Schreiber Building Room 309 at 14:30. Monday, April 2

Zemer Kosloff, Tel Aviv University

Power Weakly Mixing Transformations

**Abstract**

Schreiber Building Room 309 at 14:30.

Power Weakly Mixing Transformations

Schreiber Building Room 309 at 14:30.

Monday, April 9

Passover break - No seminar.Monday, April 16

Mikhail Sodin, Tel Aviv University

Random nodal portraits

**Abstract:**

Schreiber Building Room 309 from 14:30 to 16:00!

Random nodal portraits

We describe the progress in understanding the zero sets of smooth
Gaussian random functions of several real variables. The primary
examples are various ensembles of Gaussian real-valued polynomials
(algebraic or trigonometric) of large degree, and smooth Gaussian
functions on the Euclidean space with translation-invariant distribution.
The fundamental question is the one on the asymptotic behaviour of
the number of connected components of the zero set. This can be
viewed as a statistical version of Hilbert's 16th problem.

We start with an intriguing heuristics suggested by Bogomolny and Schmit, which relates nodal portraits of 2D Gaussian monochromatic waves to bond percolation on the square lattice. Then we explain how Ergodic Theorem and rudimentary harmonic analysis help to find the order of growth of the typical number of connected components of the zero set. We will mention a number of basic questions, which remain widely open.

The talk is based on joint works Fedor Nazarov.

Longer seminar! Note special time!We start with an intriguing heuristics suggested by Bogomolny and Schmit, which relates nodal portraits of 2D Gaussian monochromatic waves to bond percolation on the square lattice. Then we explain how Ergodic Theorem and rudimentary harmonic analysis help to find the order of growth of the typical number of connected components of the zero set. We will mention a number of basic questions, which remain widely open.

The talk is based on joint works Fedor Nazarov.

Schreiber Building Room 309 from 14:30 to 16:00!

Monday, April 23

Ross Pinsky, Technion

Probabilistic and Combinatorial Aspects of the Card-Cyclic to Random Insertion Shuffle

**Abstract**

Schreiber Building Room 309 at 14:30.

Probabilistic and Combinatorial Aspects of the Card-Cyclic to Random Insertion Shuffle

Schreiber Building Room 309 at 14:30.

Monday, April 30

Sasha Sodin, IAS, Princeton

A 2D gradient model with non-convex interaction

**Abstract:**

A 2D gradient model with non-convex interaction

The talk will be about a statistical mechanics
model on the 2D lattice. We shall define it, and
discuss the connection to height models,
the six-vertex model, and dipole gas (all of these
will be defined as well). Also, we shall discuss
the conjectured phase diagram, and explain
which parts are rigorously justified.

Based on discussions with David Brydges and Tom Spencer.

Schreiber Building Room 309 at 14:30. Based on discussions with David Brydges and Tom Spencer.

Monday, May 7

Yan V Fyodorov (Queen Mary, University of London)

Freezing Transition: from 1/f landscapes to Characteristic Polynomials of Random Matrices and the Riemann zeta-function

**Abstract:**

Freezing Transition: from 1/f landscapes to Characteristic Polynomials of Random Matrices and the Riemann zeta-function

In the talk (based on a joint work with G Hiary and J Keating; arXiv:1202.4713)
I will argue that the freezing transition scenario, previously
conjectured to take place in the statistical mechanics of 1/f-noise random energy models,
governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials of large
random unitary (CUE) matrices. I then conjecture that the results extend to the large values taken by the Riemann zeta-function over stretches of the critical line s=1/2+it of constant length, and present the results of numerical computations of the large values of $\zeta(1/2+it)$. The main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random matrix theory, and the theory of the Riemann zeta function.

Schreiber Building Room 309 at 14:30. Monday, May 14

Mark Rudelson, University of Michigan

Row products of random matrices

**Abstract:**

Row products of random matrices

We study spectral and geometric properties of a certain class of random matrices with dependent rows, which are constructed from random matrices with independent entries. For K matrices of size d times n we define the row product as a matrix of the size d^K times n, whose rows are entry-wise products of the rows of the original matrices. Such constructions arise in several computer science problems. Simulations show that, despite the dependency between the entries, these matrices behave like random matrices of the same size with independent entries. We will discuss how far this similarity can be extended.

Schreiber Building Room 309 at 14:30. Double seminar! Note special time! Each talk will be 50 minutes.

Monday, May 21

Elchanan Mossel, University of California, Berkeley

Mira Shamis, Institute for Advanced Study, Princeton

Mossel title: On reverse hypercontractive inequalities.

**Mossel Abstract:**

**Shamis Abstract:**

Mira Shamis, Institute for Advanced Study, Princeton

Mossel title: On reverse hypercontractive inequalities.

A hyper-contractive inequality for an operator T states that |Tf|_q
≤ |f|_p where q > p > 1 for all functions f. Hyper contractive
inequalities play a crucial role n discrete Fourier analysis and in
probabilistic applications. A reverse hyper-contractive inequality for
the operator T states that |Tf|_q ≥ |f|_p for q < p < 1 (q and p
can be negative) and all strictly positive functions f. The first
reverse hyper-contractive inequalities were proved by Borell more than
2 decades ago. While these inequalities may look obscure, they have
been used for the solution of a number of problems in the last decade.
I will survey applications of the inequalities and discuss new results
relating reverse hyper-contractive inequalities to hyper-contractive,
Log-Sobolev and Poincare inequalities as well as some new
applications. This is a joint work with K Oleszkiewicz (Warsaw) and A
Sen (Cambridge).

Shamis title: Anderson localization for non-monotone Schroedinger operators.
We show how the fractional moment method of Aizenman and Molchanov can be applied to a class of Anderson-type models with non-monotone potentials, to prove (spectral and dynamical) localization. The main new feature of our argument is that it does not assume any a priori Wegner-type estimate: the (nearly optimal) regularity of the density of states is established as a byproduct of the proof. The argument is applicable to finite-range alloy-type models and a class of operators with matrix-valued potentials.

(joint work with A. Elgart and S. Sodin)

Schreiber Building Room 309 at 14:10. (joint work with A. Elgart and S. Sodin)

Monday, May 28

Israel Mathematical Union meeting - No seminar.Monday, June 4

Noga Alon, Tel Aviv University

The chromatic number of random Cayley graphs

**Abstract:**

The chromatic number of random Cayley graphs

The study of random Cayley graphs of finite groups is related to the
investigation of Expanders and to problems in combinatorial Number Theory
and in Information Theory. I will discuss this topic, focusing on the
question of estimating the chromatic number of a random Cayley graph of a
given group with a prescribed number of generators. One representative
result is the fact that if p is a large prime and S is a random subset
of at most 0.5 log_2 p members of Z_p, then the chromatic number of
the Cayley graph of Z_p with respect to S is almost surely exactly 3.

Schreiber Building Room 309 at 14:30. Double seminar! Note special time and place! Each talk will be 50 minutes.

Monday, June 11

Loren Coquille, Université de Genève

Vincent Beffara, ENS Lyon

Coquille title: On the Gibbs measures of the 2d Ising and Potts models.

**Coquille Abstract:**

**Beffara Abstract:**

Vincent Beffara, ENS Lyon

Coquille title: On the Gibbs measures of the 2d Ising and Potts models.

In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the 2d Ising model are a convex combination of the two pure phases. After introducing the relevant definitions and concepts needed to understand
the physical content of this result, I will present a finite-volume, quantitative analogue (implying the classical claim). The scheme of the proof seems more natural and provides a better picture of the underlying physical phenomenon. (Joint work with Yvan Velenik).
Moreover, this new approach is substantially more robust : I will explain in parallel how it extends to the Potts model, and what are the major technical and physical difficulties to overcome. (Joint work with Hugo Duminil-Copin, Dima Ioffe and Yvan Velenik).

Beffara title: Self-interacting random walks and growth processes.
We will present a surprising link between the once-reinforced random walk, which is a self-interacting process on a lattice, and a growth system related to first-passage percolation on the same lattice. This gives a very precise description of the scaling limit of the walk, which exhibits a deterministic asymptotic shape.

Schreiber Building Room 006 at 14:10. Double seminar! Note special time! Each talk will be 50 minutes.

Monday, June 18

Nayantara Bhatnagar, University of California, Berkeley

Jonathan Hermon, Weizmann Institute

Bhatnagar title: Reconstruction on Trees.

**Bhatnagar Abstract:**

**Hermon Abstract:**

Jonathan Hermon, Weizmann Institute

Bhatnagar title: Reconstruction on Trees.

For spin systems on a tree, roughly, the reconstruction problem is to
determine whether correlations persist between vertices deep inside the
tree and the root. Reconstruction on trees plays an important role in
explaining threshold phenomena in random constraint satisfaction problems on sparse random graphs.

In the talk, I will speak about results on the threshold for reconstruction for colorings and algorithms for finding thresholds (work that is joint with Vera, Vigoda, and Weitz and with Maneva). I'll also mention results on the reconstruction threshold for independent sets (work that is joint with Sly and Tetali).

Hermon title: Social Connectivity of Random Mobile Networks.In the talk, I will speak about results on the threshold for reconstruction for colorings and algorithms for finding thresholds (work that is joint with Vera, Vigoda, and Weitz and with Maneva). I'll also mention results on the reconstruction threshold for independent sets (work that is joint with Sly and Tetali).

Joint work with Gady Kozma and Itai Benjamini - In this work we study questions related to the evolution of random mobile social networks in various graphs. In particular we study the time it takes them to become connected and for a giant component to emerge. We do so by considering a model in which there are many random walkers (which are the people of the social network) performing independent random walks on a given graph, where new acquaintances occur when walkers collide. The main reasults are polylogarithmic bounds on the social connectivity time when the underlying graph on which the walkers walk is of bounded degree. Other results concerning the complete graph (and relations to G(n,p)), expanders, the cycle and Z^{d} (and relations to percolation) might be discussed as time permits.

Schreiber Building Room 309 at 14:10. Monday, October 31

Daniel Johannsen, Tel Aviv University

The Degree Sequence of Random Planar Maps

**Abstract:**

The Degree Sequence of Random Planar Maps

In this talk we study how typical specimen of various classes of planar
maps (i.e., embedded planar graphs) look like. In particular, we are
interested in the degree sequence of a random map drawn from all maps
of equal size. For ordinary random maps, it is known that the expected
number of vertices of a fixed degree is linear in the number of edges
of that map. Moreover, this number is sharply concentrated around its
expectation for which an asymptotic formula (depending on the given
degree) exists. We see how this result can be transferred to other
classes of random maps like those that are biconnected, 3-connected,
loopless, bridgeless, or simple.

This is joint work with Konstantinos Panagiotou.

Schreiber Building Room 309 at 14:30. This is joint work with Konstantinos Panagiotou.

Monday, November 7

Ohad Feldheim, Tel Aviv University

Rigidity of 3-colorings of the d-dimensional discrete torus

**Abstract:**

Rigidity of 3-colorings of the d-dimensional discrete torus

We prove that a uniformly chosen proper coloring of
Z_{2n}^d with 3 colors has a very rigid structure when the
dimension d is sufficiently high. The coloring takes one color on
almost all of either the even or the odd sub-lattice. In
particular, one color appears on nearly half of the lattice sites.
This model is the zero temperature case of the 3-states anti-ferromagnetic Potts model, which has been studied extensively in
statistical mechanics. The proof involves results about graph
homomorphisms and various combinatorial methods, and follows a
topological intuition. Joint work with Ron Peled.

Schreiber Building Room 309 at 14:30. Monday, November 14

No seminar due to nearby conference.Monday, November 21

Ariel Yadin, Ben-Gurion University

Super-critical self avoiding walk is space filling

**Abstract:**

Super-critical self avoiding walk is space filling

The Self Avoiding Walk (SAW) is a model proposed by Flory in the 50's
to study polymer chains.
The models studied basically refer to choosing a self avoiding path, P,
in a lattice, with probability proportional to x^|P| , where |P| is the
length of P.
Here x is some positive real parameter. There is a conjectured phase
transition of the behavior of a typical path according to whether x is
above or below the "connective constant" of the lattice. The
sub-critical case has been studied by Ioffe.

In dimension 2, it is conjectured that the critical case should converge to SLE(8/3). Resolving this conjecture is a major open problem in probability. In fact not much is known about geometric properties of the 2D SAW. The super-critical SAW is expected to be space-filling, and in dimension 2 to have the scaling limit SLE(8), which is a space filling curve.

In joint work with Gady Kozma and Hugo Duminil-Copin we show that super-critical SAW is space filling. The proof is fairly natural and uses a renormalization idea. The proof works for different lattices and all dimensions > 1.

We will not assume prior knowledge of SAW or SLE.

Schreiber Building Room 309 at 14:30. In dimension 2, it is conjectured that the critical case should converge to SLE(8/3). Resolving this conjecture is a major open problem in probability. In fact not much is known about geometric properties of the 2D SAW. The super-critical SAW is expected to be space-filling, and in dimension 2 to have the scaling limit SLE(8), which is a space filling curve.

In joint work with Gady Kozma and Hugo Duminil-Copin we show that super-critical SAW is space filling. The proof is fairly natural and uses a renormalization idea. The proof works for different lattices and all dimensions > 1.

We will not assume prior knowledge of SAW or SLE.

Monday, November 28

Doron Puder, The Hebrew University of Jerusalem

Uniform Words are Primitive

**Abstract:**

Uniform Words are Primitive

Let a,b,c,... in S_n be random permutations on n elements, chosen at uniform distribution. What is the distribution of the permutation obtained by a fixed word in the letters a,b,c,..., such as ab,a^2, a^2bc^2b, or aba^(-2)b^(-1)? More concretely, do these new random permutations have uniform distribution? In general, a free word w in the free group F_k is called *uniform* if for every finite group G, the word map $w: G^k \to G$ induces uniform distribution on G (given uniform distribution on G^k). So which words are uniform?

This question is strongly connected to the notion of primitive words in the free group F_k. The word w is called*primitive* if it belongs to some basis, i.e. a free generating set. It is an easy observation that a primitive word is uniform. It was conjectured that the converse is also true. We prove it for F_2, and in a recent joint work with O. Parzanchevski, we manage to prove the conjecture in full. A key ingredient of the proofs is a new algorithm to detect primitive elements.

Schreiber Building Room 309 at 14:30. This question is strongly connected to the notion of primitive words in the free group F_k. The word w is called

Monday, December 5

Mike Hochman, The Hebrew University of Jerusalem

Equidistribution from fractals

**Abstract:**

Equidistribution from fractals

By a classical result of Cassels and W. Schmidt, the standard
middle-third Cantor sets contains normal numbers in any base which is not
a power of 3 (a number is normal in base n if it equidistributes under the
times-n mod 1 map). There are many extensions of this problem, e.g. Host's
theorem about more general times-3 invariant measures, and results of
Kauffman showing that there are many numbers with partial bounded
quotients which are normal in every base.

I will discuss a new geometric/dynamical method with which one can prove results of this kind. The method has many advantages over the analytical approach which is usually used, for example it is robust when the target set is smoothly perturbed. I will also describe applications to some of the open problems in the field. This is joint work with Pablo Shmerkin.

Schreiber Building Room 309 at 14:30. I will discuss a new geometric/dynamical method with which one can prove results of this kind. The method has many advantages over the analytical approach which is usually used, for example it is robust when the target set is smoothly perturbed. I will also describe applications to some of the open problems in the field. This is joint work with Pablo Shmerkin.

Double seminar! Note special time! Each talk will be 50 minutes.

Monday, December 12

Alexandre Stauffer, Microsoft Research

Tom Ellis, Tel Aviv University

Stauffer title: Space-time percolation and detection by mobile nodes.

**Stauffer Abstract:**

**Ellis Abstract:**

Tom Ellis, Tel Aviv University

Stauffer title: Space-time percolation and detection by mobile nodes.

Consider a Poisson point process of intensity lambda in R^d. We denote
the points as nodes and let each node move as an independent Brownian
motion. Consider a target particle that is initially placed at the origin
at time 0 and can move according to any continuous function. We say that
the target is detected at time t if there exists at least one node within
distance 1 of the target at time t. We show that if lambda is
sufficiently large, the target will eventually be detected even if
its motion can depend on the past, present and future positions of the
nodes. In the proof we use coupling and multi-scale analysis to model this event as
a fractal percolation process and show that a good event percolates in
space and time.

Ellis title: The Brownian web is a two dimensional black noise.
The Brownian web is a stochastic process constructed from Brownian motions,
and was one of the first known examples of a "black noise" in the sense of
Tsirelson. I will discuss what it means to be a black noise, and
demonstrate how the Brownian web is in fact a two-dimensional black noise.
It is only the second known example of a two-dimensional black noise after
Schramm and Smirnov's result on the scaling limit of critical planar
percolation.

Joint work with Ohad Feldheim.

Schreiber Building Room 309 at 14:10. Joint work with Ohad Feldheim.

Monday, December 19

Oren Louidor, University of California, Los Angeles

Trapping in the random conductance model.

Trapping in the random conductance model.

We consider random walks on Z^d among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but which can be arbitrarily close to zero. Our focus is on the detailed properties of the paths of the random walk conditioned
to return back to the starting point after time 2n. We show that in the situations when the heat kernel exhibits subdiffusive behavior --- which is known to be possible in dimensions d ≥ 4-- the walk gets trapped for time of order n in a small spatial region.
This proves that the strategy used to infer subdiffusive lower bounds on the heat kernel in earlier studies of this problem is in fact dominant. In addition, we settle a conjecture on the maximal possible subdiffusive decay in four dimensions and prove that
anomalous decay is a tail and thus zero-one event.

Joint work with Marek Biskup, Alexander Vandenberg and Alexander Rozinov.

Schreiber Building Room 309 at 14:30. Joint work with Marek Biskup, Alexander Vandenberg and Alexander Rozinov.

Special Joint GAFA-Horowitz seminar! Note special place and time! Each talk will be 50 minutes.

Monday, December 26

Yuval Peres, Microsoft Research

Peter Mester, Hebrew University of Jerusalem

Peres title: Mixing times are hitting times of large sets.

**Peres Abstract:**

**Mester Abstract:**

Peter Mester, Hebrew University of Jerusalem

Peres title: Mixing times are hitting times of large sets.

Consider a simple random walk on a connected, undirected, finite graph G. Mixing times and hitting times are fundamental parameters of the walk. We relate them by showing that the mixing time of a lazy walk is equivalent (up to a bounded factor) to the maximum over initial states x and large sets A of the hitting time of A starting from x. (Call a set “large” if its volume is at least 1/4 the volume of G). As an application, we show that the mixing time on a finite tree is robust to bounded change of edge conductances. It is an open problem whether the equivalence holds when we require the target set A to have volume at least Vol(G)/2. (joint work with Perla Sousi, University of Cambridge)

Mester title: A factor of i.i.d. with continuous marginals and infinite clusters spanned by identical labels.
If (G,X,\mu) and (G, Y, \nu) are systems (where G is a
group acting with measure preserving transformations on the
probability spaces (X,\mu) and (Y,\nu)), then we say that the
latter is a factor of the former if there is a map from X to Y
which commutes with the group actions and the pushforward of \mu is
\nu. We will deal with the case where both X and Y are [0,1]^G
and the measure \mu is the product measure obtained by putting the
Lebesgue measure on [0,1], factors of this system are called "factor
of i.i.d.". In a more probabilistic language we may say that we label
the group elements with i.i.d. uniform reals from [0,1] and then
use some (deterministic) equivariant rule to "relabel" the group. In
this context Gaboriau asked if the following is true: Assume that
a factor of i.i.d. has the further property that the marginal labels
are continuous (or equivalently uniform on [0,1]) and consider
the process on a Cayley graph in the group. Does it follow that
the clusters spanned by vertices of equal labels must be finite? We will
present a construction which answers this question in the negative. The
construction is a probabilistic one using simple facts about
percolation on a tree.

Shenkar Building, Holzblat Hall 007 at 14:10. Monday, January 2

Anita Winter, Universität Duisburg-Essen

The Aldous move on cladograms in the diffusion limit

**Abstract:**

The Aldous move on cladograms in the diffusion limit

A n-phylogenetic tree is a semi-labeled, unrooted and binary
tree with n leaves labeled 1,2,...,n and with (n-2) unlabeled internal
leaves and positive edge lengths representing the time spans between
common ancestors.
In biological systematics phylogenetic trees
are used to represent the evolutionary relationship between species.
If one does focus only on the kinship (that is taking all edge length of
unit length), a more precise term is cladogram.

Aldous constructed a Markov chain on cladograms and gave bounds on their mixing time. On the other hand, Aldous also gave a notion of convergence of cladograms which shows that the uniform cladogram with N leaves and edge length re-scaled by a factor of 1/sqrt{N} converges to the so-called Continuum Random Tree (CRT). These two results suggest that if we re-scale edge lengths by a factor of 1/\sqrt{N} and speeding up time by a factor of N^{3/2} the Aldous move on cladograms converges in some sense to a continuous tree-valued diffusion. We will use Dirichlet form methods to construct limiting dynamics. (This is joint work with Leonid Mytnik, Technion Haifa)

Schreiber Building Room 309 at 14:30. Aldous constructed a Markov chain on cladograms and gave bounds on their mixing time. On the other hand, Aldous also gave a notion of convergence of cladograms which shows that the uniform cladogram with N leaves and edge length re-scaled by a factor of 1/sqrt{N} converges to the so-called Continuum Random Tree (CRT). These two results suggest that if we re-scale edge lengths by a factor of 1/\sqrt{N} and speeding up time by a factor of N^{3/2} the Aldous move on cladograms converges in some sense to a continuous tree-valued diffusion. We will use Dirichlet form methods to construct limiting dynamics. (This is joint work with Leonid Mytnik, Technion Haifa)

Monday, January 9

Boris Tsirelson, Tel Aviv University

Noise as a Boolean algebra of sigma-fields

**Abstract:**

Noise as a Boolean algebra of sigma-fields

The black noise of two-dimensional percolation, disclosed recently by
Schramm, Smirnov and Garban, exceeds the limits of the existing
framework based on one-dimensional intervals. I propose another
framework --- Boolean algebras of sigma-fields.
(See also http://arXiv.org/abs/1111.7270/)

Schreiber Building Room 309 at 14:30. Monday, January 16

Subhroshekhar Ghosh, University of California, Berkeley

What does a Point Process Outside a Domain tell us about What's Inside?

**Abstract:**

What does a Point Process Outside a Domain tell us about What's Inside?

In a Poisson point process we have independence between disjoint spatial
domains, so the points outside a disk give us no information on the points
inside. The story gets a lot more interesting for processes with stronger
spatial correlation. In the case of Ginibre ensemble, a process arising
from eigenvalues of random matrices, we prove that the outside points
determine exactly the number of points inside, and further, we demonstrate
that they determine nothing more. In the case of zero ensembles of
Gaussian power series, we prove that the outside points determine exactly
the number and the centre of mass of the inside points, and nothing
further. These phenomena suggest a certain hierarchy of point processes
according to their rigidity; Poisson, Ginibre and the Gaussian power
series fit in at levels 0, 1 and 2 in this ladder.
Time permitting, we will also look at some interesting consequences
of our results, with applications to continuum percolation, reconstruction
of Gaussian entire functions, and others. This is based on joint work with
Fedor Nazarov, Yuval Peres and Mikhail Sodin.

Schreiber Building Room 309 at 14:30. Monday, January 23

Gidi Amir, Bar-Ilan University

A continuum of exponents for the rate of escape of random walks on groups

**Abstract:**

A continuum of exponents for the rate of escape of random walks on groups

For every 3/4 ≤ beta < 1 we construct a finitely generated group so that
the expected distance of the simple random walk from its starting point after n steps is n^beta (up to constants). This answers a question of Vershik, Naor and Peres. In other examples, the speed exponent can fluctuate between any two values in this interval.

Previous examples were only of exponents of the form 1-1/2^k or 1, and were based on lamplighter (wreath product) constructions. (Other than the standard beta=1/2 and beta=1 which are simply diffusive and ballistic behaviours known for a wide variety of groups) In this lecture we will describe how a variation of the lamplighter construction, namely the permutational wreath product, can be used to get precise bounds on the rate of escape in terms of return probabilities of the random walk on some Schreier graphs. We will then show how groups of automorphisms of rooted trees, related to automata groups, can then be constructed and analyzed to get the desired rate of escape. This is joint work with Balint Virag of the University of Toronto. No previous knowledge of automaton groups or wreath products is assumed.

Schreiber Building Room 309 at 14:30. Previous examples were only of exponents of the form 1-1/2^k or 1, and were based on lamplighter (wreath product) constructions. (Other than the standard beta=1/2 and beta=1 which are simply diffusive and ballistic behaviours known for a wide variety of groups) In this lecture we will describe how a variation of the lamplighter construction, namely the permutational wreath product, can be used to get precise bounds on the rate of escape in terms of return probabilities of the random walk on some Schreier graphs. We will then show how groups of automorphisms of rooted trees, related to automata groups, can then be constructed and analyzed to get the desired rate of escape. This is joint work with Balint Virag of the University of Toronto. No previous knowledge of automaton groups or wreath products is assumed.

Monday, January 30

Omer Bobrowski, Technion

Distance functions, critical points and topology for some random complexes

**Abstract:**

Distance functions, critical points and topology for some random complexes

In this talk we focus on the distance function from a random set of points P in the Euclidean space. The distance function is continuous, however, it is not everywhere differentiable. Nevertheless, one can accurately define critical points and then apply Morse theory to it.

We study the number of critical points in small neighborhoods around P. Specifically, we are interested in the case where the number of points in P goes to infinity, and the size of the neighborhoods goes to zero. We present limit theorems for the number of critical points and show that it exhibits a phase transition, depending on how fast the size of the neighborhoods goes to zero. A similar phase transition was presented recently by Kahle and Meckes who studied the Betti-numbers of random geometric complexes. We show that this is more than just a coincidence, and discuss the connection between the distance function and geometric complexes.

Schreiber Building Room 309 at 14:30. We study the number of critical points in small neighborhoods around P. Specifically, we are interested in the case where the number of points in P goes to infinity, and the size of the neighborhoods goes to zero. We present limit theorems for the number of critical points and show that it exhibits a phase transition, depending on how fast the size of the neighborhoods goes to zero. A similar phase transition was presented recently by Kahle and Meckes who studied the Betti-numbers of random geometric complexes. We show that this is more than just a coincidence, and discuss the connection between the distance function and geometric complexes.

Monday, February 21

Manjunath Krishnapur, Indian Institute of Science, Bangalore

The single ring theorem

**Abstract:**

The single ring theorem

What are the eigenvalues of a typical matrix with a given set of
singular values? We make the question precise by considering a random
matrix of the form A=UDV, where D is a diagonal matrix and U,V are
independent unitary matrices sampled from Haar measure.Under certain
assumptions on the distribution of D, we show that A has a limiting
spectral distribution, and characterize its properties. In particular,
it has the surprising feature that the support of the limit spectral
distribution is a connected annulus. This is joint work with Alice
Guionnet and Ofer Zeitouni.

Schreiber Building Room 309 at 14:30. Monday, February 28

Tom Ellis, University of Cambridge

From Diffusion Limited Aggregation to the Brownian Web via Conformal Mappings

**Abstract:**

From Diffusion Limited Aggregation to the Brownian Web via Conformal Mappings

Diffusion Limited Aggregation (DLA) is a planar growth model where the rate
of growth at at point on the boundary of a cluster is proportional to the
harmonic measure there. I will define the Hastings-Levitov growth models
which approximate DLA, and show how they use ideas of conformal invariance
which are analogous to those from the celebrated area of Schramm-Loewner
Evolution.

I will demonstrate how the evolution of the harmonic measure on the cluster boundary corresponds to a stochastic flow, and will explain why -- in a suitable scaling limit -- the flow converges to an independent system of coalescing Brownion motions, known as the Brownian web.

Schreiber Building Room 309 at 14:30. I will demonstrate how the evolution of the harmonic measure on the cluster boundary corresponds to a stochastic flow, and will explain why -- in a suitable scaling limit -- the flow converges to an independent system of coalescing Brownion motions, known as the Brownian web.

Monday, March 7

Nati Linial, Hebrew University

Probability and topology? What a strange combination

**Abstract:**

Probability and topology? What a strange combination

In this talk I review several recent attempts at introducing tools from
the probabilistic method to the study of certain topological
structures. This includes joint papers and ongoing work with: Roy
Meshulam, Mishael Rosenthal, Tahl Nowik, Lior Aronshtam and Tomasz
Luczak.

Schreiber Building Room 309 at 14:30. Monday, March 14

Michael Bromberg, Tel Aviv University

Weak invariance principle for the local times of partial sums of Markov Chains

**Abstract**

Schreiber Building Room 209 at 14:30.

NOTE SPECIAL ROOM!

Weak invariance principle for the local times of partial sums of Markov Chains

Schreiber Building Room 209 at 14:30.

NOTE SPECIAL ROOM!

Monday, March 21

Omri Sarig, Weizmann Institute and Pennsylvania State University

Symbolic dynamics for C^{1+epsilon} surface diffeomorphisms with positive topological entropy

**Abstract:**

Symbolic dynamics for C^{1+epsilon} surface diffeomorphisms with positive topological entropy

Suppose f:M -->M is a C^{1+epsilon} diffeomorphism of a compact
smooth manifold of dimension two with topological entropy h>0.
For every 0<delta<h, we construct a "delta-large" invariant set E
such that f restricted to E has a countable Markov partition. It
follows that f|E is a finite-to-one factor of a topological Markov
shift.
"Delta-large" means that E has full measure for every ergodic invariant
measure with entropy bigger than delta.
There are many consequences, for example -- every ergodic measure of
maximal entropy is a finite-to-one factor of a positive recurrent
countable Markov chain, and is therefore isomorphic to a Bernoulli
scheme times a rotation.

Schreiber Building Room 309 at 14:30. Monday, March 28

Elon Lindenstrauss, Hebrew University

Orbit closures for some higher rank Abelian actions

**Abstract:**

Orbit closures for some higher rank Abelian actions

In 1967 Furstenberg discovered that any x2, x3 closed
invariant subset of R/Z is closed or finite, contrasting with the
situation for sets invariant under a single endomorphism. Even
earlier, Cassels and Swinnerton-Dyer have made a deep conjecture on
products of linear forms that is equivalent to a rigidity statement
about orbit closures of the diagonal group in SL(n,R)/SL(n,Z) (namely
that they are compact iff the orbit is periodic).
However, the exact classification of orbit closures can be quite
tricky, even on the conjectural level. I will present some positive
and negative results in this direction based mostly on joint works
with U. Shapira and Z. Wang.

Schreiber Building Room 309 at 14:30. Monday, April 4

Johan Tykesson, Weizmann Institute

Percolation in the vacant set of Poisson cylinders

**Abstract:**

NOTE SPECIAL ROOM!

Percolation in the vacant set of Poisson cylinders

We consider a Poisson point process on the space of lines in R^d, where
a multiplicative factor u>0 of the intensity measure determines the
density of lines. Each line in the process is taken as the axis of a
bi-infinite cylinder of radius 1. We investigate percolative properties
of the vacant set, defined as the subset of R^d that is not covered by
any such cylinder. We show that in dimensions d >= 4, there is a
critical value u_*(d) \in (0,\infty), such that with probability 1, the
vacant set has an unbounded component if u<u_*(d), and only bounded
components if u>u_*(d). For d=3, we prove that the vacant set does
not percolate for large u and that the vacant set intersected with a
two-dimensional subspace of R^d does not even percolate for small
u>0.
This is joint work with David Windisch.

Schreiber Building Room 210 at 14:30. NOTE SPECIAL ROOM!

Monday, April 11

Haya Kaspi, Technion

Stochastic PDE limits of many servers queues

**Abstract**

Schreiber Building Room 309 at 14:30.

Stochastic PDE limits of many servers queues

Schreiber Building Room 309 at 14:30.

Monday, April 18

Passover break - No seminar.Monday, April 25

Passover break - No seminar.Monday, May 2

Eviatar Procaccia, Weizmann Institute

Geometry of the Random interlacement

**Abstract:**

Geometry of the Random interlacement

We consider the geometry of random interlacements on the
$d$-dimensional lattice. We use ideas from stochastic dimension theory
proved in \cite{benjamini2004geometry} to prove the following: Given
that two vertices $x,y$ belong to the interlacement set, it is possible
to find a path between $x$ and $y$ contained in the trace left by at
most $\lceil d/2 \rceil$ trajectories. Moreover, this result is sharp
in the sense that there are pairs of points in the interlacement set
which cannot be connected by a path using the traces of at most $\lceil
d/2 \rceil-1$ trajectories.

Schreiber Building Room 309 at 14:30. Monday, May 9

Memorial day - No seminar.Monday, May 16

Ran Tessler, Hebrew University

Zero Temperature Spin Glass Models

**Abstract:**

Zero Temperature Spin Glass Models

In this introductory talk we describe the Edwards-Anderson-Ising Spin
Glass model. We consider ground states of this model on several types
of graphs. In addition we describe a new dynamical process, which is a
natural generalization of the famous Glauber dynamics (in zero
temperature) whose limits are supported on the set of ground-states.

Joint work with Noam Berger.

Schreiber Building Room 309 at 14:30. Joint work with Noam Berger.

Monday, May 23

Joanna Kulaga, Nicolaus Copernicus University, Torun, Poland

Joining primeness property of order n

**Abstract:**

Joining primeness property of order n

Joining primeness property of order n (JP(n)) is a generalization of
such well-known properties as e.g. minimal self joining or simplicity.
JP(n) property is also connected with singularity of the so-called
maximal spectral type. On the other end of the "map" of possible
dynamical systems lies a class of infinitely divisible systems. During
my talk I will talk about the relation between these notions. As a
byproduct I sketch a simple proof of Girsanov's theorem about spectral
multiplicity function for Gaussian systems.

Schreiber Building Room 309 at 14:30. Monday, May 30

Boris Solomyak, University of Washington

Tiling dynamical systems with an infinite invariant measure

**Abstract:**

Tiling dynamical systems with an infinite invariant measure

We investigate non-primitive, non-minimal tiling substitutions in R^d,
and characterize all "natural" invariant measures for the corresponding
dynamical systems. In many cases, such an invariant measure is unique
in some sense, and it is infinite (but sigma-finite). The first such
example, in 1 dimension ("integer Cantor set") was considered by Alby
Fisher. Our examples include the "Sierpinski gasket" and "Sierpinski
carpet" tilings in the plane. (Joint work with Maria Isabel Cortez.)

Schreiber Building Room 309 at 14:30. Monday, June 6

Isaac Meilijson, Tel Aviv University

Granule membranes play dice

**Abstract:**

NOTE SPECIAL ROOM!

Granule membranes play dice

Granules are secreted by the cell in either spontaneous or evoked mode,
and the distribution of their volume is recorded, which we have modeled
respectively as the exit and stationary distributions of a Growth &
Elimination Markovian stochastic model. Growth requires granule-granule
fusion and elimination requires granule-membrane fusion, by linking
SNARE protein pairs diffusing in the granule and membrane surfaces.
Transition rates of the Growth & Elimination model depend on the
required number of such pairs for fusion, that can be identified
(theoretically and experimentally estimated) from the two secretion
volume distributions. Based on joint work with Ilan Hammel and Eyal
Nitzany.

Schreiber Building Room 209 at 14:30. NOTE SPECIAL ROOM!

Monday, October 18

Ron Peled, Tel Aviv University

An Introduction to Statistical Mechanics and Phase Transition Phenomenon

**Abstract:**

An Introduction to Statistical Mechanics and Phase Transition Phenomenon

We will give an introduction to several basic models in statistical
mechanics, including percolation, Ising and random surface models, and
explain the phase transitions they exhibit. No previous knowledge in
statistical mechanics will be assumed.

Schreiber Building Room 209 at 14:30. Monday, October 25

Gady Kozma, Weizmann Institute

Random walks on the symmetric group

**Abstract:**

Random walks on the symmetric group

The group of all permutations of n elements is perhaps the most studied
finite non-abelian group, in terms of random walk on it. An exciting
aspect of the field is that every question can be approached
probabilistically or algebraically, and people reached impressive
results both using and not using its theory of representations. We will
survey old and new results, and connections to the quantum Heisenberg
ferromagnet. No prior knowledge of representation theory (or physics)
will be assumed.

Schreiber Building Room 209 at 14:30. Monday, November 1

Nicholas Crawford, Technion

A Curious Example of Localization via Randomness in Classical Statistical Mechanics

**Abstract**

Schreiber Building Room 309 at 14:30.

A Curious Example of Localization via Randomness in Classical Statistical Mechanics

Schreiber Building Room 309 at 14:30.

Monday, November 8

Senya Shlosman, Centre de Physique Théorique, Marseille, France

Understanding the arctic circle in the six-vertex model, and related open questions

**Abstract:**

Understanding the arctic circle in the six-vertex model, and related open questions

I will review some properties of the six-vertex model with domain-wall
boundary conditions, its relation with alternating sign matrices and
totally symmetric plane partitions. I will then present some results
concerning the arctic circle properties in this model (joint results
with Pavel Bleher). Finally I will formulate some open questions
concerning the behavior of non-intersecting random walks on the plane.

Schreiber Building Room 309 at 14:30. Monday, November 15

Barak Weiss, Ben-Gurion University

On separated nets

**Abstract:**

On separated nets

A separated net is a uniformly discrete subset of R^d, i.e. any
point in R^d is bounded distance from a point in the set, and there is a
lower bound on distance of points in the set from each other. One
way to construct dynamical separated nets is to consider return
times to a section of a minimal R^d-flow on a compact space. Gromov asked
whether any separated net is bilipschitz to Z^d, and a counterexample was
given in 1998 by Burago, Kleiner and McMullen. However for natural
dynamical separated nets, it is in general a hard question to decide
whether they are bilipschitz to Z^d. I will present some recent results
on dynamical separated nets, joint with Alan Haynes and Yaar Solomon.

Schreiber Building Room 309 at 14:30. Monday, November 22

Ofer Zeitouni, Weizmann Institute

From branching random walks to Gaussian Free Fields

**Abstract:**

From branching random walks to Gaussian Free Fields

The (discrete) Gaussian Free Field on a finite graph is the process
{X_v} with density proportional to e^{-\sum_{v~w} (X_v-X_w)^2};
it has played an important role in many aspects of contemporary probability
theory, as well as in mathematical physics through models for random
interfaces and quantum gravity. A natural question relates to the magnitude of
fluctuations of the maximum of the field; the planar case, where the graph is
a box of side N, is of particular interest.

Branching random walks (BRW's) model the (spatial) evolution of a population, where particles split and then perform independent random motion (on R). As was shown by Bramson in the late 1970's, the behavior of the outliers of the population (ie, the particles that moved farthest from the starting point, after n generations) is determined, in the Gaussian displacement case, by solutions of the Kolmogorov-Petrovsky-Piscounov equation. In the non-Gaussian case, a proof that fluctuations of the maximum are of order 1 was given only very recently.

I will describe surprising links that BRW's have with Gaussian free fields, first passage percolation, and the cover time of graphs by random walks; I will then explain how arguments developed for the study of both Gaussian and non-Gaussian BRW's played a role in a recent resolution of the conjecture that fluctuations of the maximum of the Gaussian Free Field in dimension 2 are bounded.

Based on joint works with E. Bolthausen, J.-D. Deuschel and with M. Bramson

Schreiber Building Room 309 at 14:30. Branching random walks (BRW's) model the (spatial) evolution of a population, where particles split and then perform independent random motion (on R). As was shown by Bramson in the late 1970's, the behavior of the outliers of the population (ie, the particles that moved farthest from the starting point, after n generations) is determined, in the Gaussian displacement case, by solutions of the Kolmogorov-Petrovsky-Piscounov equation. In the non-Gaussian case, a proof that fluctuations of the maximum are of order 1 was given only very recently.

I will describe surprising links that BRW's have with Gaussian free fields, first passage percolation, and the cover time of graphs by random walks; I will then explain how arguments developed for the study of both Gaussian and non-Gaussian BRW's played a role in a recent resolution of the conjecture that fluctuations of the maximum of the Gaussian Free Field in dimension 2 are bounded.

Based on joint works with E. Bolthausen, J.-D. Deuschel and with M. Bramson

Monday, November 29

Svetlana Katok, Penn State University

Structure of attractors for (a,b)-continued fraction transformations

**Abstract:**

Structure of attractors for (a,b)-continued fraction transformations

I will discuss a two-parameter family of one-dimensional maps related
to so-called (a,b)-continued fractions and studied jointly with Ilie
Ugarcovici. The associated natural extension maps have attractors with
finite rectangular structure for the entire parameter set except for a
Cantor-like set of one-dimensional Lebesgue zero measure on the
boundary that we completely described. The structure of these
attractors can be “computed” from the data (a, b), and for a dense open
set of parameters the Reduction theory conjecture holds, i.e. every
point is mapped to the attractor after finitely many iterations. If
time permits, I will explain how this theory can be applied to the
study of ergodic properties of associated Gauss-like maps and for
coding of geodesics on the modular surface.

Schreiber Building Room 309 at 14:30. Monday, December 6

Anatole Katok, Penn State University

Absolutely continuous invariant measure for commuting maps and flows

**Abstract:**

Absolutely continuous invariant measure for commuting maps and flows

The only situation In traditional dynamics where an open set of a
system admits an absolutely continuous invariant measure appears for
expanding maps of the circle. For invertible systems, i.e.
diffeomorphisms and smooth flows, existence of such a measure is at
best an infinite codimension condition. For smooth systems with
multi-dimensional time, i.e. commuting maps or vector fields, the
situation changes dramatically.
Here certain dynamical properties of an invariant measure involving
Lyapunov characteristic exponents and entropy
imply absolute continuity. As an application certain global topological
conditions on an action, such as homotopy types of its elements, imply
existence of an absolutely continuous invariant measure.

Schreiber Building Room 309 at 14:30. Monday, December 13

Ori Gurel-Gurevich, University of British Columbia, Vancouver, Canada

Poisson Thickening

**Abstract:**

Poisson Thickening

Can a Poisson process be thickened?
That is, can more points be added deterministically to a Poisson process,
so that the resulting process is also a Poisson process (of higher intensity)?
We will show that this can be done, but not equivariantly
(i.e. not in a way that commutes with some shift).

In recent years, there has been much interest in problems of this kind: given a stochastic spatial process X, can it be extended to another process Y (perhaps under additional constraints)? For example, can the cells of a Poisson-Voronoi tessellation be colored deterministically and equivariantly, such that adjacent cells have different colors?

We will survey results of this kind, with particular emphasis on those which yield pretty pictures and explain the solution to the thickening problem in some detail.

Joint Work with Ron Peled.

Schreiber Building Room 309 at 14:30. In recent years, there has been much interest in problems of this kind: given a stochastic spatial process X, can it be extended to another process Y (perhaps under additional constraints)? For example, can the cells of a Poisson-Voronoi tessellation be colored deterministically and equivariantly, such that adjacent cells have different colors?

We will survey results of this kind, with particular emphasis on those which yield pretty pictures and explain the solution to the thickening problem in some detail.

Joint Work with Ron Peled.

Monday, December 20

Yuval Peres, Microsoft Research

Brownian motion with variable drift and the Pascal principle

**Abstract:**

Brownian motion with variable drift and the Pascal principle

It is well known that Brownian motion on the line has no isolated zeros
almost surely; Does this hold if we add a deterministic variable drift?

Similarly, we know that planar Brownian motion has paths of zero area almost surely; does this hold when adding a variable drift? Does the expected volume of the Wiener sausage increase when we add drift?

We will answer these questions and relate the last one to random geometric graphs, rearrangement inequalities and the “Pascal principle”.

(Talk based on joint works with T. Antunovic, J. Miller, J. Ruscher, P. Sousi, A. Stauffer, A. Sinclair and B. Vermesi).

Schreiber Building Room 309 at 14:30. Similarly, we know that planar Brownian motion has paths of zero area almost surely; does this hold when adding a variable drift? Does the expected volume of the Wiener sausage increase when we add drift?

We will answer these questions and relate the last one to random geometric graphs, rearrangement inequalities and the “Pascal principle”.

(Talk based on joint works with T. Antunovic, J. Miller, J. Ruscher, P. Sousi, A. Stauffer, A. Sinclair and B. Vermesi).

Monday, December 27

Eitan Bachmat, Ben-Gurion University

A symmetry satisfied by the Pollaczek - Khinchine formula

**Abstract:**

A symmetry satisfied by the Pollaczek - Khinchine formula

Queueing theory is an important branch of applied probability. A
fundamental result in queueing theory is the Pollaczek-Khinchine
formula (1930, 1932) for the average waiting time in a single server
queue with Poisson arrivals. We provide new insight regarding this old
formula by showing that it has a certain symmetry/duality.
Interestingly, this type of symmetry first appeared in a simple
computation of Riemann (1859). The symmetry is useful in the analysis
of queues which are typical of supermarkets or minimarkets. If time
permits we will explain how these might be improved in practice,
saving! a few precious minutes of our lives.

Schreiber Building Room 309 at 14:30. Monday, January 3

Hugo Duminil-Copin, Université de Genève

Critical temperature of the square lattice Potts model

**Abstract**

Schreiber Building Room 309 at 14:30.

Critical temperature of the square lattice Potts model

Schreiber Building Room 309 at 14:30.

Monday, January 10

Ivan Corwin, Courant Institute, New York University

Beyond the Gaussian Universality Class

**Abstract:**

Beyond the Gaussian Universality Class

The Gaussian central limit theorem says that for a wide class of
stochastic systems, the bell curve (Gaussian distribution) describes
the statistics for random fluctuations of important observables. In
this talk I will look beyond this class of systems to a collection of
probabilistic models which include random growth models, polymers,
particle systems, matrices and stochastic PDEs, as well as certain
asymptotic problems in combinatorics and representation theory. I will
explain in what ways these different examples all fall into a single
new universality class with a much richer mathematical structure than
that of the Gaussian.

Schreiber Building Room 309 at 14:30. Monday, January 17

David Ralston, Ben-Gurion University

Heavy Sets: Structure and Dimension

**Abstract:**

Heavy Sets: Structure and Dimension

Given a measure preserving transformation on a space and a function of
zero mean, the heavy set is the set of points whose ergodic sums remain
nonnegative for all (forward) time. After discussing a few generic
results, we will turn our attention to the specific setting where the
transformation is an irrational circle rotation and the function is the
characteristic function of the interval [0,1/2]. In this setting, we
will show that the heavy set is almost-surely (with regard to the
rotation parameter) the union of a perfect Cantor set of both Hausdorff
and box dimension c, where c is some constant strictly between zero and
one, together with countably many isolated points.

Schreiber Building Room 309 at 14:30. Past years

School of Mathematical Sciences