Dominic Yeo, Technion
Frozen percolation with k types
Abstract:
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:
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.
Monday, May 22
Barak Weiss, Tel Aviv University
Random walks on homogeneous spaces and diophantine approximation on fractals
Abstract:
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:
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:
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.
Monday, June 12
Matthew Kwan, ETH Zürich
Random designs
Abstract:
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:
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.
Monday, June 26
Omer Tamuz, Caltech
Large deviations in social learning
Abstract:
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.
Fall Semester 2016
Monday, October 31
Xiaolin Zeng, Tel Aviv University
Recurrence of two dimensional edge-reinforced random walk
Abstract:
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:
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:
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:
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:
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:
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:
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.
Monday, December 19
Percy Deift, Courant Institute, New York University
Universality in numerical computations with random data. Analytical results
Abstract:
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:
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.
Monday, January 9
Daniel El-Baz, Tel Aviv University
Angles in adelic quasicrystals and related problems
Abstract:
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:
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:
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.
Spring Semester 2016
Monday, February 1
Special semester break seminar!
Gaultier Lambert, KTH University, Stockholm
Mesoscopic linear statistics of determinantal processes
Abstract:
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.
Monday, February 29
Mikhail Sodin, Tel Aviv University
Spectral measures of {0, 1}-stationary sequences
Abstract:
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:
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.
Monday, March 14
Aser Cortines, Technion
Finite-size corrections to the speed of a branching-selection process
Abstract:
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.
Monday, March 21
Oren Louidor, Technion
Aging in a logarithmically correlated potential
Abstract:
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:
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 walks
Glazman Abstract:
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.
Monday, April 4
Amos Nevo, Technion
Hyperbolic geometry and pointwise ergodic theorems
Abstract:
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:
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 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:
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.
Monday, May 16
Alexander Shamov, Weizmann Institute
Introduction to Gaussian multiplicative chaos
Abstract:
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:
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.
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:
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
Feldheim Abstract:
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.
Monday, June 6
Brandon Seward, Hebrew University of Jerusalem
The Ornstein isomorphism theorem for countably infinite groups
Abstract:
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.
Fall Semester 2015
Monday, October 19
Ryokichi Tanaka, Tohoku University
Random walks on hyperbolic groups: entropy and speed
Abstract:
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.
Monday, October 26
Asaf Nachmias, Tel Aviv University
Indistinguishability of trees in uniform spanning forests
Abstract:
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.
Monday, November 2
Nishant Chandgotia, Tel Aviv University
Entropy Minimality and Four-Cycle Free Graphs
Abstract:
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:
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:
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:
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:
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:
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:
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:
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:
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.
Monday, January 4
Omri Sarig, Weizmann Institute
Measures of maximal entropy for surface diffeomorphisms (joint with Buzzi and Crovisier)
Abstract:
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 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.
Special Summer Seminar 2015
Note special room and time!
Monday, July 20
Frank den Hollander, Leiden University
Breaking of ensemble equivalence in complex networks
Abstract:
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.
Spring Semester 2015
Monday, March 9
Omer Angel, University of British Columbia
Increasing subsequences in random walks
Abstract:
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:
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:
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.
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:
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.
Monday, April 20
Benjamin Weiss, The Hebrew University of Jerusalem
Weak mixing properties for non-singular actions
Abstract:
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:
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.
Monday, May 4
Mira Shamis, Weizmann Institue
Wegner estimates for deformed Gaussian ensembles, and the Wegner orbital model
Abstract:
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.
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.
Note special time!
Monday, May 18
Jonathan Hermon, UC Berkeley
Characterization of cutoff for reversible Markov chains
Abstract:
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.
Monday, May 25
Margherita Disertori, Universität Bonn
Some results on history dependent stochastic processes
Abstract:
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 8
Yuval Peres, Microsoft Research
Using random walks to analyze Prediction with Expert Advice
Abstract:
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.
Monday, June 15
Ohad Feldheim, University of Minnesota
Long-range order in random three colourings of Zd
Abstract:
Consider a random colouring of a bounded domain in Zd 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.
Fall Semester 2014
Monday, October 27
Uri Shapira, Technion
Escape of mass for measures invariant under the diagonal group
Abstract:
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:
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.
Monday, November 10
Antal Jarai, University of Bath
Sandpiles and the one-end property of uniform spanning forests
Abstract:
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:
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:
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.
Monday, December 1
Eitan Bachmat, Ben-Gurion University
A geometric Pollaczek-Khinchine formula involving increasing subsequences, modular transformations and optics
Abstract:
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:
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.
Monday, December 15
Daniel Ueltschi, Warwick
Random loop models and quantum spin systems
Abstract:
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:
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:
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 Edges
Harel Abstract:
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:
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:
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:
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.
Special Summer Seminar 2014
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 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.
Spring Semester 2014
Monday, February 17
Elliot Paquette, Weizmann Institute
Choices and Intervals
Abstract:
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.
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:
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
Marchese Abstract:
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:
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.
Monday, March 10
Mike Hochman, The Hebrew University of Jerusalem
Dimension of self-similar sets and additive combinatorics
Abstract:
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:
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.
Monday, March 24
Idan Perl, Ben-Gurion University
Extinction window of mean field branching annihilating random walk
Abstract:
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:
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
Manolescu Abstract:
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:
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:
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 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:
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:
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:
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.
Monday, June 9
Eliran Subag, Weizmann Institute
Extremal point processes and freezing
Abstract:
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.
Fall Semester 2013
Monday, October 14
Isaac Meilijson, Tel Aviv University
Placing multiple chips in the roulette table
Abstract:
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:
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.
Monday, October 28
Alexander Gorodnik, University of Bristol
Diophantine approximation by orbits
Abstract:
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:
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.
Monday, November 11
Omer Tamuz, Microsoft Research and MIT
Stabilizer Rigidity in Irreducible Group Actions
Abstract:
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.
Monday, November 18
Naomi Feldheim, Tel Aviv University
Long gaps between sign-changes of stationary Gaussian processes
Abstract:
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.
Monday, November 25
Roman Kotecký, University of Warwick and Charles University
Long range order for random colourings on planar lattices
Abstract:
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.
Monday, December 2
Manfred Einsiedler, ETH Zurich
Equidistribution and p-adic dynamics
Abstract:
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:
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.
Monday, December 16
Dima Ioffe, Technion
Low temperature interfaces, layering transitions and Ferrari-Spohn diffusions
Abstract:
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.
Monday, December 23
Ariel Yadin, Ben Gurion University
Towards an inverse Kleiner Theorem for finitely generated groups
Abstract:
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.
Monday, December 30
Boris Solomyak, University of Washington
On the Fourier asymptotics of self-similar measures
Abstract:
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:
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
Merrill Abstract:
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.
Schreiber Building Room 309 at 14:10.
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:
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 11
Cyrille Lucas, Weizmann Institute
Uniform Internal Diffusion Limited Aggregation
Abstract:
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.
Monday, April 8
Ronen Eldan, Weizmann Institute
On the connection between the spectral gap of convex bodies and the variance conjecture.
Abstract:
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 29
Gidi Amir, Bar-Ilan University
Multiple excited random walk, excited Mob and leftover environments
Abstract:
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.