Horowitz Seminar on Probability, Ergodic Theory and Dynamical Systems

Monday, March 5

Barak Weiss, Ben-Gurion University
The Mordell-Gruber spectrum and homogeneous dynamics.
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, March 12

Noam Berger, The Hebrew University of Jerusalem
New conditions for ballisticity for RWRE
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, March 19

Agelos Georgakopoulos, Technical University, Graz
Random walks, electrical networks, and the role of the Poisson boundary
Abstract:

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

Schreiber Building Room 309 at 14:30.

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

Monday, April 16

Mikhail Sodin, Tel Aviv University
Random nodal portraits
Abstract:

We describe the progress in understanding the zero sets of smooth Gaussian random functions of several real variables. The primary examples are various ensembles of Gaussian real-valued polynomials (algebraic or trigonometric) of large degree, and smooth Gaussian functions on the Euclidean space with translation-invariant distribution. The fundamental question is the one on the asymptotic behaviour of the number of connected components of the zero set. This can be viewed as a statistical version of Hilbert's 16th problem.
We start with an intriguing heuristics suggested by Bogomolny and Schmit, which relates nodal portraits of 2D Gaussian monochromatic waves to bond percolation on the square lattice. Then we explain how Ergodic Theorem and rudimentary harmonic analysis help to find the order of growth of the typical number of connected components of the zero set. We will mention a number of basic questions, which remain widely open.
The talk is based on joint works Fedor Nazarov.

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

Monday, April 30

Sasha Sodin, IAS, Princeton
A 2D gradient model with non-convex interaction
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, May 7

Yan V Fyodorov (Queen Mary, University of London)
Freezing Transition: from 1/f landscapes to Characteristic Polynomials of Random Matrices and the Riemann zeta-function
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, May 14

Mark Rudelson, University of Michigan
Row products of random matrices
Abstract:

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

Schreiber Building Room 309 at 14:30.

Double seminar! Note special time! Each talk will be 50 minutes.
Monday, May 21

Elchanan Mossel, University of California, Berkeley
Mira Shamis, Institute for Advanced Study, Princeton
Mossel title: On reverse hypercontractive inequalities.
Mossel Abstract:

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

Shamis title: Anderson localization for non-monotone Schroedinger operators.
Shamis Abstract:

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

Schreiber Building Room 309 at 14:10.

Fall Semester 2011

Monday, October 31

Daniel Johannsen, Tel Aviv University
The Degree Sequence of Random Planar Maps
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, November 7

Ohad Feldheim, Tel Aviv University
Rigidity of 3-colorings of the d-dimensional discrete torus
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, November 21

Ariel Yadin, Ben-Gurion University
Super-critical self avoiding walk is space filling
Abstract:

The Self Avoiding Walk (SAW) is a model proposed by Flory in the 50's to study polymer chains. The models studied basically refer to choosing a self avoiding path, P, in a lattice, with probability proportional to x^|P| , where |P| is the length of P. Here x is some positive real parameter. There is a conjectured phase transition of the behavior of a typical path according to whether x is above or below the "connective constant" of the lattice. The sub-critical case has been studied by Ioffe.
In dimension 2, it is conjectured that the critical case should converge to SLE(8/3). Resolving this conjecture is a major open problem in probability. In fact not much is known about geometric properties of the 2D SAW. The super-critical SAW is expected to be space-filling, and in dimension 2 to have the scaling limit SLE(8), which is a space filling curve.
In joint work with Gady Kozma and Hugo Duminil-Copin we show that super-critical SAW is space filling. The proof is fairly natural and uses a renormalization idea. The proof works for different lattices and all dimensions > 1.
We will not assume prior knowledge of SAW or SLE.

Schreiber Building Room 309 at 14:30.

Monday, November 28

Doron Puder, The Hebrew University of Jerusalem
Uniform Words are Primitive
Abstract:

Let a,b,c,... in S_n be random permutations on n elements, chosen at uniform distribution. What is the distribution of the permutation obtained by a fixed word in the letters a,b,c,..., such as ab,a^2, a^2bc^2b, or aba^(-2)b^(-1)? More concretely, do these new random permutations have uniform distribution? In general, a free word w in the free group F_k is called uniform if for every finite group G, the word map $w: G^k \to G$ induces uniform distribution on G (given uniform distribution on G^k). So which words are uniform?
This question is strongly connected to the notion of primitive words in the free group F_k. The word w is called primitive if it belongs to some basis, i.e. a free generating set. It is an easy observation that a primitive word is uniform. It was conjectured that the converse is also true. We prove it for F_2, and in a recent joint work with O. Parzanchevski, we manage to prove the conjecture in full. A key ingredient of the proofs is a new algorithm to detect primitive elements.

Schreiber Building Room 309 at 14:30.

Monday, December 5

Mike Hochman, The Hebrew University of Jerusalem
Equidistribution from fractals
Abstract:

By a classical result of Cassels and W. Schmidt, the standard middle-third Cantor sets contains normal numbers in any base which is not a power of 3 (a number is normal in base n if it equidistributes under the times-n mod 1 map). There are many extensions of this problem, e.g. Host's theorem about more general times-3 invariant measures, and results of Kauffman showing that there are many numbers with partial bounded quotients which are normal in every base.
I will discuss a new geometric/dynamical method with which one can prove results of this kind. The method has many advantages over the analytical approach which is usually used, for example it is robust when the target set is smoothly perturbed. I will also describe applications to some of the open problems in the field. This is joint work with Pablo Shmerkin.

Schreiber Building Room 309 at 14:30.

Double seminar! Note special time! Each talk will be 50 minutes.
Monday, December 12

Alexandre Stauffer, Microsoft Research
Tom Ellis, Tel Aviv University
Stauffer title: Space-time percolation and detection by mobile nodes.
Stauffer Abstract:

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

Ellis title: The Brownian web is a two dimensional black noise.
Ellis Abstract:

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

Schreiber Building Room 309 at 14:10.

Monday, December 19

Oren Louidor, University of California, Los Angeles
Trapping in the random conductance model.

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

Schreiber Building Room 309 at 14:30.

Special Joint GAFA-Horowitz seminar! Note special place and time! Each talk will be 50 minutes.
Monday, December 26

Yuval Peres, Microsoft Research
Peter Mester, Hebrew University of Jerusalem
Peres title: Mixing times are hitting times of large sets.
Peres Abstract:

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

Mester title: A factor of i.i.d. with continuous marginals and infinite clusters spanned by identical labels.
Mester Abstract:

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

Shenkar Building, Holzblat Hall 007 at 14:10.

Monday, January 2

Anita Winter, Universität Duisburg-Essen
The Aldous move on cladograms in the diffusion limit
Abstract:

A n-phylogenetic tree is a semi-labeled, unrooted and binary tree with n leaves labeled 1,2,...,n and with (n-2) unlabeled internal leaves and positive edge lengths representing the time spans between common ancestors. In biological systematics phylogenetic trees are used to represent the evolutionary relationship between species. If one does focus only on the kinship (that is taking all edge length of unit length), a more precise term is cladogram.
Aldous constructed a Markov chain on cladograms and gave bounds on their mixing time. On the other hand, Aldous also gave a notion of convergence of cladograms which shows that the uniform cladogram with N leaves and edge length re-scaled by a factor of 1/sqrt{N} converges to the so-called Continuum Random Tree (CRT). These two results suggest that if we re-scale edge lengths by a factor of 1/\sqrt{N} and speeding up time by a factor of N^{3/2} the Aldous move on cladograms converges in some sense to a continuous tree-valued diffusion. We will use Dirichlet form methods to construct limiting dynamics. (This is joint work with Leonid Mytnik, Technion Haifa)

Schreiber Building Room 309 at 14:30.

Monday, January 9

Boris Tsirelson, Tel Aviv University
Noise as a Boolean algebra of sigma-fields
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, January 16

Subhroshekhar Ghosh, University of California, Berkeley
What does a Point Process Outside a Domain tell us about What's Inside?
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, January 23

Gidi Amir, Bar-Ilan University
A continuum of exponents for the rate of escape of random walks on groups
Abstract:

For every 3/4 ≤ beta < 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point after n steps is n^beta (up to constants). This answers a question of Vershik, Naor and Peres. In other examples, the speed exponent can fluctuate between any two values in this interval.
Previous examples were only of exponents of the form 1-1/2^k or 1, and were based on lamplighter (wreath product) constructions. (Other than the standard beta=1/2 and beta=1 which are simply diffusive and ballistic behaviours known for a wide variety of groups) In this lecture we will describe how a variation of the lamplighter construction, namely the permutational wreath product, can be used to get precise bounds on the rate of escape in terms of return probabilities of the random walk on some Schreier graphs. We will then show how groups of automorphisms of rooted trees, related to automata groups, can then be constructed and analyzed to get the desired rate of escape. This is joint work with Balint Virag of the University of Toronto. No previous knowledge of automaton groups or wreath products is assumed.

Schreiber Building Room 309 at 14:30.

Monday, January 30

Omer Bobrowski, Technion
Distance functions, critical points and topology for some random complexes
Abstract:

In this talk we focus on the distance function from a random set of points P in the Euclidean space. The distance function is continuous, however, it is not everywhere differentiable. Nevertheless, one can accurately define critical points and then apply Morse theory to it.
We study the number of critical points in small neighborhoods around P. Specifically, we are interested in the case where the number of points in P goes to infinity, and the size of the neighborhoods goes to zero. We present limit theorems for the number of critical points and show that it exhibits a phase transition, depending on how fast the size of the neighborhoods goes to zero. A similar phase transition was presented recently by Kahle and Meckes who studied the Betti-numbers of random geometric complexes. We show that this is more than just a coincidence, and discuss the connection between the distance function and geometric complexes.

Schreiber Building Room 309 at 14:30.

Spring Semester 2011

Monday, February 21

Manjunath Krishnapur, Indian Institute of Science, Bangalore
The single ring theorem
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, February 28

Tom Ellis, University of Cambridge
From Diffusion Limited Aggregation to the Brownian Web via Conformal Mappings
Abstract:

Diffusion Limited Aggregation (DLA) is a planar growth model where the rate of growth at at point on the boundary of a cluster is proportional to the harmonic measure there. I will define the Hastings-Levitov growth models which approximate DLA, and show how they use ideas of conformal invariance which are analogous to those from the celebrated area of Schramm-Loewner Evolution.
I will demonstrate how the evolution of the harmonic measure on the cluster boundary corresponds to a stochastic flow, and will explain why -- in a suitable scaling limit -- the flow converges to an independent system of coalescing Brownion motions, known as the Brownian web.

Schreiber Building Room 309 at 14:30.

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

Monday, March 21

Omri Sarig, Weizmann Institute and Pennsylvania State University
Symbolic dynamics for C^{1+epsilon} surface diffeomorphisms with positive topological entropy
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, March 28

Elon Lindenstrauss, Hebrew University
Orbit closures for some higher rank Abelian actions
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, April 4

Johan Tykesson, Weizmann Institute
Percolation in the vacant set of Poisson cylinders
Abstract:

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

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

Monday, May 2

Eviatar Procaccia, Weizmann Institute
Geometry of the Random interlacement
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, May 16

Ran Tessler, Hebrew University
Zero Temperature Spin Glass Models
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, May 23

Joanna Kulaga, Nicolaus Copernicus University, Torun, Poland
Joining primeness property of order n
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, May 30

Boris Solomyak, University of Washington
Tiling dynamical systems with an infinite invariant measure
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, June 6

Isaac Meilijson, Tel Aviv University
Granule membranes play dice
Abstract:

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

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

Fall Semester 2010

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

Monday, October 25

Gady Kozma, Weizmann Institute
Random walks on the symmetric group
Abstract:

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

Schreiber Building Room 209 at 14:30.

Monday, November 8

Senya Shlosman, Centre de Physique Théorique, Marseille, France
Understanding the arctic circle in the six-vertex model, and related open questions
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, November 15

Barak Weiss, Ben-Gurion University
On separated nets
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, November 22

Ofer Zeitouni, Weizmann Institute
From branching random walks to Gaussian Free Fields
Abstract:

The (discrete) Gaussian Free Field on a finite graph is the process {X_v} with density proportional to e^{-\sum_{v~w} (X_v-X_w)^2}; it has played an important role in many aspects of contemporary probability theory, as well as in mathematical physics through models for random interfaces and quantum gravity. A natural question relates to the magnitude of fluctuations of the maximum of the field; the planar case, where the graph is a box of side N, is of particular interest.
Branching random walks (BRW's) model the (spatial) evolution of a population, where particles split and then perform independent random motion (on R). As was shown by Bramson in the late 1970's, the behavior of the outliers of the population (ie, the particles that moved farthest from the starting point, after n generations) is determined, in the Gaussian displacement case, by solutions of the Kolmogorov-Petrovsky-Piscounov equation. In the non-Gaussian case, a proof that fluctuations of the maximum are of order 1 was given only very recently.
I will describe surprising links that BRW's have with Gaussian free fields, first passage percolation, and the cover time of graphs by random walks; I will then explain how arguments developed for the study of both Gaussian and non-Gaussian BRW's played a role in a recent resolution of the conjecture that fluctuations of the maximum of the Gaussian Free Field in dimension 2 are bounded.
Based on joint works with E. Bolthausen, J.-D. Deuschel and with M. Bramson

Schreiber Building Room 309 at 14:30.

Monday, November 29

Svetlana Katok, Penn State University
Structure of attractors for (a,b)-continued fraction transformations
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, December 6

Anatole Katok, Penn State University
Absolutely continuous invariant measure for commuting maps and flows
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, December 13

Ori Gurel-Gurevich, University of British Columbia, Vancouver, Canada
Poisson Thickening
Abstract:

Can a Poisson process be thickened? That is, can more points be added deterministically to a Poisson process, so that the resulting process is also a Poisson process (of higher intensity)? We will show that this can be done, but not equivariantly (i.e. not in a way that commutes with some shift).
In recent years, there has been much interest in problems of this kind: given a stochastic spatial process X, can it be extended to another process Y (perhaps under additional constraints)? For example, can the cells of a Poisson-Voronoi tessellation be colored deterministically and equivariantly, such that adjacent cells have different colors?
We will survey results of this kind, with particular emphasis on those which yield pretty pictures and explain the solution to the thickening problem in some detail.
Joint Work with Ron Peled.

Schreiber Building Room 309 at 14:30.

Monday, December 20

Yuval Peres, Microsoft Research
Brownian motion with variable drift and the Pascal principle
Abstract:

It is well known that Brownian motion on the line has no isolated zeros almost surely; Does this hold if we add a deterministic variable drift?
Similarly, we know that planar Brownian motion has paths of zero area almost surely; does this hold when adding a variable drift? Does the expected volume of the Wiener sausage increase when we add drift?
We will answer these questions and relate the last one to random geometric graphs, rearrangement inequalities and the “Pascal principle”.
(Talk based on joint works with T. Antunovic, J. Miller, J. Ruscher, P. Sousi, A. Stauffer, A. Sinclair and B. Vermesi).

Schreiber Building Room 309 at 14:30.

Monday, December 27

Eitan Bachmat, Ben-Gurion University
A symmetry satisfied by the Pollaczek - Khinchine formula
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, January 10

Ivan Corwin, Courant Institute, New York University
Beyond the Gaussian Universality Class
Abstract:

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

Schreiber Building Room 309 at 14:30.

Monday, January 17

David Ralston, Ben-Gurion University
Heavy Sets: Structure and Dimension
Abstract:

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

Schreiber Building Room 309 at 14:30.

Horowitz Seminar on Probability, Ergodic Theory and Dynamical Systems

Spring Semester 2012

Fall Semester 2011

Spring Semester 2011

Fall Semester 2010