Toronto Probability Seminar

Past talks

The Schramm-Loewner evolution (SLE) is a family of random planar curves that appear as scaling limits of curves derived from a range of discrete lattice models from statistical physics. SLE is constructed by solving the Loewner differential equation with a Brownian motion as the so-called Loewner driving function. A first step towards proving convergence to SLE is to show that the Loewner driving function for the discrete curve converges to Brownian motion. In the talk we will discuss recent work on how to estimate the convergence rate to the SLE curve, given as input a convergence rate for the Loewner driving function.

Monday, May 30, 2011

Mark Holmes (Auckland)

Random walks in degenerate random environments

Abstract

In joint work with Tom Salisbury, we study random walks in i.i.d. random environments in Z^d in dimensions 2 and higher. In our environments, at any given site some steps may not be available to the random walker (i.e. we don't assume ellipticity). Among our main results are 0-1 laws for directional transience (extending results already known under the assumption of ellipticity) and a simple monotonicity result in for 2-valued environments (at each site the environment takes one of two values).

The law of the exponential functional of Lévy processes plays a prominent role from both theoretical and applied perspectives. We start this talk by describing some reasons motivating its study and we review all known results concerning the distribution of this random variable. We proceed by describing a new factorization identity for the law of the exponential functional under very mild con- ditions on the underlying Lévy process. As by-product, we provide some interesting distributional properties enjoyed by this random variable as well as some new analytical expressions for its distribution (Joint work with J.C. Pardo (CIMAT, Mexico) and P. Patie (Université Libre de Bruxelles, Belgium)).

In this talk, I will discuss the two matrix model in random matrix theory and present some recent results on the asymptotic behavior of the eigenvalues statistics. In particular, a variational problem will be introduced that characterizes the limiting eigenvalue density for one of the matrices, in case one of the potentials is quartic. I will also discuss the eigenvalue correlations at the local scale and introduce a new universality class near a multicritical point in the quartic/quadratic case.

Second-order freeness extends free probabilistic approaches to large random matrices from moments to fluctuations. Similarly to the first-order case, a definition of (complex) second-order freeness satisfied by independent ensembles of many important matrix models (Ginibre, Wishart, unitarily invariant, Haar-distributed unitary) can be used as a rule for calculating fluctuations of these matrices. However, the real analogues of these matrix models do not generally satisfy this defintion. I will examine the differences between the real and complex ensembles which appear in some of the combinatorial tools applied to these matrices, in particular the genus expansion, and present a definition for real second-order freeness satisfied by the real matrix models.

We add a random bulk term, modeling the interaction with the impurities of the medium to a standard functional in the gradient theory of phase transitions consisting of a gradient term with a double well potential. We study the existence and properties of minimizers.

The results strongly depend on dimensions and on the strength of the random field. In d bigger or equal than 3 if the strength of the random field is small enough there are a.s with respect to the random field two minimizers and we compute the surface tension of the interface. In dimensions d <3 we show that there exists only one minimizer and therefore no interfaces.

Joint work with Nicolas Dirr.

We study a model of very asymmetric directed polymers in a random environment. We compute the free energy of the model and the order of fluctuation of the partition function. As in the very asymmetric last passage percolation, the key point is an approximation by a Brownian percolation model, which has strong connections with random matrices.

The McKean-Vlasov system and its porous medium generalizations will be described. These are non--linear diffusion equations with an additional non--local non--linearity provided by convolution. Recently popular in a variety of applications, these enjoy an ancient heritage as a basis for understanding equilibrium and near equilibrium fluids. The model is discussed in finite volume where, on the basis of the physical considerations, the correct scaling (for the model itself) is identified. For dimension two and above and in large volume, various dynamical anomalies are related to phase transitions; the phase structure of the model is completely elucidated.

In the early '80s, Brox and Rost introduced the so-called Boltzmann-Gibbs principle. As an application, they deduced the time evolution of equilibrium fluctuations of the density for interacting particle systems. In one dimension, we introduce two generalizations of this principle, which we named second-order and local Boltzmann-Gibbs principle. As applications of these generalizations, we prove that equilibrium fluctuations of weakly asymmetric particle systems are given by energy solutions of the KPZ equation, and we obtain novel functional limit theorems for additive functionals of particle systems.

Joint with Patricia Gonçalves (U. do Minho-Portugal)

The theory of graphed equivalence relations provides a natural point of view on random graphs. Namely, any invariant probability measure on a graphed equivalence relation can be considered as a probability measure on the space of rooted graphs. Benjamini and Schramm showed that any weak limit of uniform measures on finite graphs is an invariant measure on the space of rooted graphs with respect to its natural equivalence relation. Recently Elek proved that, conversely, any invariant measure on the space of rooted trees can be obtained in this way. We shall show that this approximation property also holds for any invariant measure on the space of rooted graphs such that a.e. graph is Liouville (has no non-constant harmonic functions).

The environment viewed from the particle has been a powerful tool in the investigation of random conductance models. For non-reversible random walks in random environment the problem of the equivalence of the static and dynamic points of view is understood only in a few cases. The case of Dirichlet environment, which corresponds to the case where the transition probabilities at each site are iid Dirichlet random variables, is particularly interesting since its annealed law corresponds to the law of a reinforced random walk. In this talk, we will characterize, for Dirichlet environments in dimension larger or equal to 3, the cases where the static and dynamic points of view are equivalent. We can deduce from this a complete characterization of the ballistic regimes in dimension larger or equal to 3. The proof is based on crucial property of statistical invariance by time reversal valid for the class of Dirichlet environments.

I will talk about variational problems related to the stochastic Hamilton-Jacobi equations and its discrete analogues. Some of these models have laws of large numbers and for them we study the bounds on the variance and the large deviation events.

Consider the sequence of partial sums of a centered random walk with finite variance. We study asymptotics of the probability that the first $n$ terms of this sequence are positive, as $n \to \infty$. The first result here is due to Ya. Sinai (1992) who came to the problem considering solutions of the Burgers equation with random initial data. The speaker's original motivation emerged as these probabilities appeared in his study of certain properties of sticky particle systems with random initial positions. We present our results and discuss the more general problem of finding small deviation probabilities of integrated stochastic processes.

We shall discuss several conjectures regarding the spectral properties of random band matrices, and some results that can be proved using perturbation series.

A ball is moving at constant speed in straight line inside a domain D of R^d, and bounces randomly upon hitting the boundary. The sequence of impacts on th boundary is a natural random walk on the boundary of D. For general bounded domains the walk is ergodic. For the reflection law, the cosine density is of particular interest, since the uniform measure on the boundary is invariant. We consider also the case of unbounded domain, precisely the case when D is an infinite "random tube". Under general assumptions, the process is then diffusive in dimension d=3,4,... The proof uses techniques from random media.

Joint work with Sergei Popov, Gunter Schutz, Marina Vachkovskaia.

I will explain the concept of Busemann functions in Last Passage Percolation, and in particular for the generalized Hammersley process. These Busemann functions turn out to be very useful, for example to calculate asymptotic speeds of multiple second class particles in an arbitrary rarefaction intitial condition, but in this talk I will focus on how the Busemann might be used to prove cube root fluctuations. For the classical Hammersley process we have made these methods rigorous.

Finite (or fixed) rank perturbations of large random matrices arise in a number of applications. The main phenomenon is a phase transition in the largest eigenvalues as a function of the strength of the perturbation. I will describe recent and forthcoming work, joint with Balint Virag, in which we introduce a new way to study such matrices. The main idea is a reduction to a new band-diagonal form and the convergence of this form to a continuum random Schroedinger operator on the half-line. We describe the near-critical fluctuations in several ways, solving a well-known open problem in the real case. Another consequence is a new route to the Painleve structure in the celebrated Tracy-Widom distributions.

We consider random walks in an i.i.d. non-negative potential on the d-dimensional integer lattice. The walks are conditioned to hit a remote location and are studied under the annealed path measure. When the potential is bounded away from zero, it is very simple to show that the expected time needed by the conditioned random walk to reach a remote location, call it y, grows at most linearly in |y|. The question becomes much harder if the potential is allowed to be zero with positive probability. We prove that even in this situation the expected time to reach y increases only linearly in |y|. In dimension one we can show the existence of the asymptotic speed as y goes to infinity.

The motivation for this question comes from an attempt to compare Lyapunov exponents and, thus, quenched and annealed large deviations rate functions for random walks in small potentials, that are not bounded away from zero.

This is a joint work with Thomas Mountford (EPFL, Lausanne).

We consider a two-dimensional infinitesimal continuum percolation model with columnar dependence. It is related to oriented percolation, first-passage percolation, Lipschitz percolation, Poisson matchings and coverings of the circle by random arcs. Several open questions are posed.

We present a low temperature anisotropic anti-ferromagnetic 2D Ising model through the guise of a certain dimer model. This model also has a bijection with a one-dimensional particle system equipped with creation and annihilation. We can find the exact phase diagram, which determines two significant values (the independent and critical value). We also highlight some of the behavior of the model in the scaling window at criticality and at independence.

We will describe the asymptotic behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term and the long time behavior of corresponding diffusion processes. In particular, we discuss the exit problem and metastability for the processes corresponding to quasi-linear initial-boundary value problems.

M is a Meixner probability on R if for X and Y independent with distribution M and if S=X+Y then the conditional expectation of X*2 knowing S is a quadratic polynomial in S. There are 6 types of them: Bernoulli, Poisson, negative binomial, Gaussian, gamma and hyperbolic. In this lecture we consider the same problem when M is a probability on the (n,n) symmetric matrices -or more generally on Hermitian complex or quaternionic - invariant by rotation. We find back the six types again. For instance the Bernoulli type is obtained as the mixing of the distributions Mk for k=0,1,...,n where Mk is the law of the orthogonal projection on a uniformly distributed random subspace of dimension k. The Laplace transforms of these Meixner distributions are characterized by a linear system of PDE with a finite dimensional set of solutions.

This is joint work with W. Bryc.

For two-dimensional independent percolation, Russo-Seymour-Welsh (RSW) bounds on crossing probabilities are an important a-priori indication of scale invariance, and they turned out to be a key tool to describe the phase transition: what happens at and near criticality.

In this talk, we prove RSW-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. A central tool in our proof is Smirnov's fermionic observable for the FK Ising model, that makes some harmonicity appear on the discrete level, providing precise estimates on boundary connection probabilities.

We also prove several related results - including some new ones - among which the fact that there is no magnetization at criticality, tightness properties for the interfaces, and the value of the half-plane one-arm exponent.

This is joint work with H. Duminil-Copin and C. Hongler.

We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class ``close'' to a fixed (background) metric.

We explain how to estimate the probability that Gauss curvature will change sign after a random conformal perturbation of a metric; discuss some extremal problems for that probability, and their relation to other extremal problems in spectral geometry.

Generalizations to higher dimensions will be discussed in my talk at the Workshop on Geometric Probability and Optimal Transportation on Wednesday, November 3, in Room 230, 2:10 - 3:00.

This is joint work with Y. Canzani and I. Wigman

In the talk, I plan to discuss our recent results with Fedja Nazarov (arXiv:1005.4113, arXiv:1003.4251): close to optimal conditions on a test-function that yield asymptotic normality of the corresponding linear statistics of random complex zeroes; universal local bounds for k-point functions of zeroes, and their strong clustering.

We consider a branching-selection system in $\rr$ with $N$ particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as $N\to\infty$, the empirical measure process associated to the system converges in distribution to a deterministic measure-valued process whose densities solve a free boundary integro-differential equation. We also show that this equation has a unique traveling wave solution traveling at speed $c$ or no such solution depending on whether $c\geq a$ or $c

Joint work with Rick Durrett.

A biased random walker in open space will move at positive velocity. If the medium is disordered, however, the motion may be slowed to vanishing velocity by the walker encountering large connected structures in the disorder that acts as traps. A natural model for these effects is a walker on an infinite Galton-Watson tree with leaves, with a constant bias away from the root. Here, the finite trees hanging off the backbone act as traps. The progress of the walker is determined on all time-scales by a discrete inhomogeneity, in which trap sojourn times tend to cluster around powers of the bias parameter. This prevents the existence of a scaling limit. I will introduce an alternative model, in which biases on edges of the tree are randomized with a non-lattice distribution, so that a stable limiting law results. These two effects, of persistent discrete inhomogeneity in a constant bias model, and stable limiting laws in the randomly biased case, may have counterparts in more physical models in Euclidean space, where the persistent discreteness may arise as a rational resonance in the bias slope.

Coauthors: Alex Fribergh (Z^d and tree models), Gerard Ben Arous and Nina Gantert (tree models).

The 1+1 dimensional directed polymer model is a Gibbs measure on simple random walk paths of a prescribed length. The weights for the measure are determined by a random environment occupying space-time lattice sites, and the measure favors paths to which the environment gives high energy. For each inverse temperature $\beta$ the polymer is said to be in the weak disorder regime if the environment has little effect on it, and the strong disorder regime otherwise. In dimension 1+1 it turns out that all positive $\beta$ are in the strong disorder regime. I will introduce a new regime called intermediate disorder, which is accessed by scaling the inverse temperature to zero with the length $n$ of the polymer. The precise scaling is $\beta n^{-1/4}$. The most interesting result is that under this scaling the polymer has diffusive fluctuations, but the fluctuations themselves are not Gaussian. Instead they are still coupled to the random environment, and their distribution is intimately related to the Tracy-Widom distribution for the largest eigenvalue of a random matrix from the GUE. More recent work also indicates that we can take a scaling limit of the entire intermediate disorder regime to construct a continuous random path under the effect of a continuum random environment. We call the scaling limit the continuum random polymer. I will discuss a few properties of the continuum random polymer and its intimate connection to the stochastic heat equation in one dimension.

Joint work with Kostya Khanin and Jeremy Quastel.

The Hermitian matrix model is usually analyzed by a Riemann-Hilbert problem of size higher than 2. If the external source is spiked, i.e., only finitely many eigenvalues of the external source matrix are nonzero, we show a new approach to solve the problem. First we solve the rank 1 case by steepest-descent method, and then by a determinantal formula we derive the result in the higher rank case from that in the rank 1 case. We show the asymptotic behavior of the largest eigenvalue. Joint work with Jinho Baik.

We consider the weakly asymmetric limit of simple exclusion with drift to the left, starting with step Bernoulli initial data so that macroscopically one has a rarefaction fan. We study the fluctuations of the associated height function process observed along slopes in the fan, which are given by the Hopf-Cole solution of the Kardar-Parisi-Zhang equation, with appropriate initial data. Slopes strictly inside the fan correspond with Dirac delta function initial data, while at the edge of the rarefaction fan, the initial data is one sided Brownian. We provide exact formulas for the one point distributions of these KPZ fluctuations which, as time goes to infinity, recover the expected Tracy-Widom type limit.

Excited random walks (or random walks in cookie environments) are random walk models in which the probability of departing to the right from a site depends on the number of visits to that site (equivalently how many cookies have been eaten at that site) up to the present time. Work of Zerner, Basdevant and Singh, and others, has demonstrated various different types of behaviour possible with a bounded number of biased cookies at each site

We'll discuss a deterministic combinatorial result that compares two integer sequences l(n) and r(n) defined in terms of collections L and R of arrows satisfying a natural monotonicity relation. Applying this when the arrows are random, the sequences produce non-monotone couplings of self-interacting random walks (such as excited random walks in one dimension). This allows us to extend known results about such models.

For probabilists, a Gaussian spin glass is essentially a Gaussian process on {-1,+1}^N whose variance is N. An important example is the Sherrington-Kirkpatrick model whose covar$ function of the Hamming distance on {-1,+1}^N. In this talk, I will give an overview of some of the conjectures coming from physics about the behavior of the extrema in the limit of large N, as$ recent progress on these conjectures in probability. I will focus especially on the ultrametricity conjecture, which is a bold statement about the Gibbs measure of these process$ extremal statistics in general.

Click here for a pdf version of the abstract

The Wick formula reduces the computation of Gaussian integrals over a real or complex vector space to a combinatorial rule plus knowledge of a scalar matrix (which probabilists call the covariance matrix and physicists call the propagator). Interesting applications of the Wick formula often deal with Gaussian integrals over the space of Hermitian matrices, and the associated combinatorics is that of maps on surfaces. Now suppose that one wants to compute integrals over a compact group of matrices, such as the unitary group. There is an analogue of the Wick formula available in this setting - now the "propagator" is the Gram matrix associated to a certain projection. The resulting combinatorics is that of symmetric polynomials evaluated at Jucys-Murphy elements, and this connection allows one to obtain interesting asymptotics, recursive structure, character expansion, and other features of unitary matrix integrals.

We study the model of Directed Polymers in Random Environment in 1+1 dimensions, where the environment is i.i.d. with a site distribution having a polynomial tail with power -\alpha, where \alpha \in (0,2). After proper scaling of temperature 1/\beta, we show strong localization of the polymer to an optimal region in the environment where energy and entropy are best balanced. We prove that this region has a weak limit under linear scaling and identify the limiting distribution as an (\alpha, \beta)-indexed family of measures on Lipschitz curves lying inside the 45 degree rotated square with unit diagonal. In particular, this shows order of n for the transversal fluctuations of the polymer. If (and only if) \alpha is small enough, we find that there exists a random critical temperature above which the effect of the environment is not macroscopically noticeable.

Joint work with Oren Louidor.

In the talk I will present a family of new, unsolved problems and some partial solutions. The common theme is local (Benjamini-Schramm) convergence of sequences of finite graphs of bounded degree. We will use some notions of asymptotic group theory, ergodic theory and topology.

We will give an overview of the recent results regarding random matrices exhibiting local (microscopic) Poisson eigenvalue statistics. It is known that the Poisson statistics holds for certain classes of random Hermitian matrices, random unitary matrices, random operators on rooted tree graphs and it is conjectured that it also holds for random non-Hermitian Anderson models. We discuss in detail the case of the random CMV matrices, which are the unitary analog of the one-dimensional random Schrodinger operator.

Let T be the return time to the origin of a simple random walk on a recurrent graph. We show that T is heavy tailed and non-concentrated. More precisely, we have

i) P(T>t) > c/sqrt(t)
ii) P(T=t|T>=t) < C log(t)/t

Inequality i) is attained on Z, and we construct an example demonstrating the sharpness of ii). We use this example to answer negatively a question of Peres and Krishnapur about recurrent graphs with the finite collision property (that is, two independent SRW on them collide only finitely many times, almost surely).

Joint work with Asaf Nachmias.

We consider a polymer with configuration modeled by the trajectory of a Markov chain, interacting with a potential of form u+V_n when it visits a particular state 0 at time n, with V_n representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned by the potential. The main question of interest is, how does this depinning transition differ from the one in the annealed model, where the interaction is effectively homogeneous in n? The most important element is the distribution of the return time to 0 for the underlying Markov chain, and one would like to characterize those return-time distributions which do and do not result in different critical exponents, or critical points, for the quenched vs. annealed model. This problem is much better understood than it was 5 years ago, though many questions remain open. We will give an overview of some recent results.

We study random walks on the uniform spanning tree (UST) on Z^2. We obtain estimates for the transition probabilities of the random walk, the distance of the walk from its starting point after n steps, and exit times of both Euclidean balls and balls in the intrinsic graph metric. In particular, we prove that the spectral dimension of the uniform spanning tree on Z^2 is 16/13 almost surely.

In order to prove these results, we use the work of Barlow, Jarai, Kumagai, Misumi and Slade on random walks on random graphs which implies that it suffices to establish volume and effective resistance bounds for the UST. Using Wilson's algorithm, we show that this reduces to obtaining estimates on the number of steps of loop-erased random walks (LERW) in subsets of Z^2. If we let M_n be the number of steps of a LERW from the origin to the circle of radius n, then Kenyon showed that E[M_n] is logarithmically asymptotic to n^{5/4}. In addition to this fact, we need to show that with high probability, M_n is close to its mean. In fact, we will obtain exponential moment bounds for M_n which implies that the tails of M_n decay exponentially.

Joint work with Martin Barlow

Simple trends in high-dimensional data sets are modeled by spiked real Gaussian sample covariance matrices. In the large-size limit, the behaviour of the top eigenvalue exhibits a phase transition as a function of the strength of the trend. Using a stochastic operator limit approach, we show that the top eigenvalue has an asymptotic distribution near the phase transition and give several characterizations of the limit laws; one of these involves only a linear boundary value problem. In the well-studied complex case, our PDE description reproduces known explicit formulas and yields a simple new derivation of the Painleve formula for the Tracy-Widom distribution.

Joint work with Balint Virag.

The real Ginibre random matrix (the nxn matrix with all entries iid standard normals) is perhaps the most natural non-Hermitian matrix ensemble one could ask for. Its spectrum however does not enjoy the determinantal nature possessed by its complex analog, making the determination of various spectral limit laws more difficult. We show that, as expected from the complex case, the spectral radius has a classical Gumbel distribution limit. More interesting (maybe) we prove a limit law for the largest real eigenvalue, described in terms of a Fredholm determinant whose kernel (sadly) lacks some of the important structure (in particular, integrability) that one is accustomed to in random matrix theory.

Joint work with Christopher Sinclair.

We complete a result originally due to Evans and O'Connell in 1994 which equates the law of a supercritical, super-diffusion with quadratic branching mechanism to that of the spatial tree of a particle branching particle diffusion "dressed" with immigrating subcritical superdiffusions. We give a direct pathwise construction allowing for a fully general branching mechanism. The key to our construction is the use of the Dynkin-Kuznetsov N-measures. This is based on joint work with Julien Berestycki and Antoni Murillo.

Random symmetric matrices with entries that vanish outside a band around the diagonal, but otherwise have independent identically distributed matrix elements, were introduced in the physics literature as an effective model of a "localization/delocalization" transition seen in disordered materials. In this talk, it will be shown that such matrices satisfy a localization condition which guarantees that eigenvectors have strong overlap with only Wµ standard basis vectors where W is the band width and µ is an positive exponent. This statement is vacuous if Wµ > N, the size of the matrix, but if Wµ << N as N tends to infinity, then a typical eigenvector is essentially supported on a vanishing fraction of standard basis vectors. For a Gaussian band ensemble, with matrix elements given by i.i.d. centered Gaussians within a band of width W, the estimate µ ~ ln 8 holds. The role of this band matrix model in physics and some open problems and conjectures will also be discussed.

It is an open problem to determine the number of infinite-volume ground states in the Edwards-Anderson (nearest neighbor) spin glass model on Z^d for d \geq 2 (with, say mean zero Gaussian couplings). This is a limiting case of the problem of determining the number of extremal Gibbs states at low temperature. In both cases, there are competing conjectures for d \geq 3, but no complete results even for d=2. I report on new results which go some way toward proving that (with zero external field, so that ground states come in pairs, related by a global spin flip) there is only a single ground state pair (GSP). Our result is weaker in two ways: First, it applies not to the full plane Z^2, but to a half-plane. Second, rather than showing that a.s. (with respect to the quenched random coupling realization J) there is a single GSP, we show that there is a natural joint distribution on J and GSP's such that for a.e. J, the conditional distribution on GSP's given J is supported on only a single GSP. The methods used are a combination of percolation-like geometric arguments with translation invariance (in one of the two coordinate directions of the half-plane) and uses as a main tool the ``excitation metastate'' which is a probability measure on GSP's and on how they change as one or more individual couplings vary.

Joint work with Louis-Pierre Arguin, Michael Damron, and Dan Stein.

Wiener-Hopf factorization and related fluctuation identities allow us to study various functionals of a Levy process, such as extrema, first/last passage time, overshoot and undershoot, etc. Due to many applications of Levy processes in Mathematical Finance there is a lot of interest in studying processes for which one can compute distributions of these functionals analytically. Unfortunately the list of such examples is very short: it contains processes with one-sided jumps, subclass of stable processes, processes with hyperexponential jumps. In this talk we will discuss several recent results on Wiener-Hopf factorization for processes having meromorphic characteristic exponent. This class is a natural generalization of processes having rational transform; it shares many key properties related to the analytical structure of Wiener-Hopf factorization, but at the same time it is a much larger class which allows for more flexible modeling of jumps.

A perfect matching on 2n is an n disjoint paring of 2n numbers. Given a matching, one can consider crossings and nestings. For example, the number of perfect matchings with no crossings equal the Catalan number, and so is the number of matchings with no nestings. Recently, Chen, Deng, Du, Stanley and Yan studies various properties of the maximal crossings and maximal nestings of a matching. Especially they are shown to be symmetric random variables if we consider uniformly random matchings.

It was also deduced, by making connection to various existing works, that the maximal crossings and the maximal nestings are asymptotically distributed as the GOE Tracy-Widom distribution from random matrix theory. In this talk, we will show that they are asymptotically independent. We use the determinantal formula obtained by the above authors. Connections to an integrable differential equation, the Ablowitz-Ladik equation, will also be discussed.

Joint work with Robert Jenkins.

In this lecture we will discuss a phase transition in the Liouville property for automaton groups.

We will start by discussing the Liouville property , which says that the only bounded harmonic functions on a graph are constant, and connect it to amenability of groups. We will then describe automaton groups - which have played an important role in providing (counter-)examples to many problems in group theory, (e.g. groups with intemediate growth) and their Schreier graphs represent classical fractals such as the Sierpinski gasket and Julia sets of finite polynomials such as the Basilica set.

Our main result shows that there is a phase transition in the Liouville property of automaton groups with respect to a certain classification - the degree of activity growth - that will be described. We conclude that all automaton groups with up to linear activity are amenable. The proof has many probablistic elements such as random walks on the Schreier graphs and electrical networks.

This talk is based on joint work with O. Angel and B. Virag

For large unitary matrices, the number of eigenvalues in distinct shrinking intervals satisfies a central limit theorem, whose covariance structure is related to some branching processes. In this talk, we present these random matrix results, following works of Diaconis, Evans and Wieand, and show the strict analogue for the zeros of L-functions.

Recent progress in the theory of hyperbolic billiards has enhanced the interest towards billiard models since they are the most promising for the derivation of laws of statistical physics from newtonian dynamics. Parallel to this progress, energy transfer models (in particular, the study of diffusion and of Fourier's law of heat conduction) have recently come to the center of interest. In this talk I will present two determinstic models with energy transfer and will treat a stochastic version of one of them in detail.

In bond percolation on the simple cubic lattice, each bond is independently "open" with probability p. Suppose we view each open bond as a solid but flexible bar, with all bars that share an endpoint being joined at that point. Then it is possible for two disjoint connected components to be topologically entangled. Is it possible for all connected components to be finite, and yet for an infinite number of them to form a single entangled cluster? G. Grimmett an A. Holroyd showed that this happens (almost surely) for some values of p, but not when p is very close to 0. They then asked whether the number of entangled clusters (modulo translation) with exactly N edges is bounded exponentially in N. We prove that the answer is yes. Among our corollaries we obtain

(1) an improved lower bound on the critical value for this "entanglement percolation", and

(2) exponential decay of the tail probabilities for the size of the entangled cluster containing the origin, when p is small.

This is joint work with Mahshid Atapour.

There are two natural ways of producing repelling random point-clouds on the complex plane: either by taking the eigenvalues of some random matrix or by considering the zeros of a random (Gaussian) analytic function. I will explain some old and new connections between the two worlds.

At each vertex of the planar square lattice, independently lay down pairs of ar$ (with probability 1-p). Then do a simple random walk according to the available$ the results obtained is that for p close to 1/2 there is an infinite cluster of$ are finite.

Joint work with Mark Holmes (Auckland).

The voter model can be seen as a simple model for describing the propagation of opinions in a population where neighbors influence each other. More precisely, every integer is assigned with an original opinion at time t=0 and then updates its opinion by taking on the opinion of one of its neighbors chosen uniformly at random with rate 1. In the first part of the talk, I will show that such a model can easily be described in terms of a system of coalescing random walks. In the second part of the talk, I will introduce a variation of the preceding model where the voters do not only change their mind under the influence of their environment, but where they are also able to come up with an opinion differing from their neighbors. This model is closely related to a classical model in statistical physics called the one dimensional stochastic Potts model. I will show that under the appropriate scaling, this model converges to a continuum object which can be constructed by a marking procedure of a family of coalescing Brownian motions.

Joint work with C. Newman and K. Ravishankar.

Start with n particles at the origin in Z^2, and let each perform a simple random walk until it reaches an unoccupied site. Lawler, Bramson and Griffeath proved that with high probability the resulting set of n occupied sites is close to a disk. The order of fluctuations from circularity remains an open problem. I'll describe a way of modifying slightly the law of the walk so that the limiting shape becomes a diamond instead of a disk. There is a natural one-parameter family of walks of this type, which exhibit a phase transition in the order of fluctuations.

Joint work with Wouter Kager.

We show that for all $p > p_c(\Z^d)$ percolation parameters, the probability that the cluster of the origin is finite but is adjacent to the infinite cluster with at least $t$ edges is exponentially small in $t$. This result yields a simple proof that the isoperimetric profile of the infinite cluster basically coincides with the profile of the original lattice, which implies that simple random walk on the cluster behaves the same way. The same results hold for all finitely presented groups if $p$ is close enough to 1, but renormalization can be used on $\Z^d$ to get the full result.

We also examine the possibility of renormalization on other groups. Itai Benjamini conjectured that if a group $G$ is scale-invariant in the sense that has a finite index subgroup chain $G = G_0 >  G_1 >  G_2 > \dots$ with $G_i\simeq G$ and $\bigcap_i G_i=\{1\}$, then it has to be of polynomial growth. In joint work with V. Nekrashevych, we have given several counterexamples: the lamplighter group $\Z_2 \wr \Z$, the solvable Baumslag-Solitar groups $BS(1,m)$, and the affine groups $\Z^d \rtimes GL(\Z,d)$ are all scale-invariant.

We shall discuss the adaptive dynamics of predator prey systems modeled by a dynamical system in which the characteristics are allowed to evolve by small random mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit prey coexistence and the parameters of the resident prey type converge to the solution of an ODE. When only the predators are allowed to evolve, coexistence of predators occurs. Depending on the parameters being varied we see (i) the number of coexisting predators remains tight and the differences of the parameters from a reference species converge in distribution to a limit, or (ii) the number of coexisting predators tends to infinity and we can study the evolving process of coexisting predator characteristics via connections with killed branching random walks and a Brunet-Derrida type branching-selection particle system.

We obtain the large N asymptotics of the partition function of the six vertex model of statistical physics with domain wall boundary conditions. The solution is based on the Zinn-Justin's reduction of the partition function to a random matrix model integral and on the Riemann-Hilbert approach.

Given a Poison point process of unit masses (stars) in dimension d >= 3, Newtonian gravity partitions space into domains of attraction (cells) of equal volume. The allocation is translation equivariant - the shape of cells do not depend on absolute position in space. We investigate the quantitative geometry of the allocation's cells. Our first result shows that a.s. all cells are bounded and that their diameters have exponential tails. We continue and investigate large deviations for the cell. More precisely, we find that the probability that mass exp(R^t) in a cell travels distance R decays like exp(R^{f_d(t)}) where we identify the functions f_d exactly. These functions are piecewise smooth and the discontinuities of f_d represent phase transitions. In dimension d = 3, the large deviation is due to a distant attracting galaxy but a phase transition occurs when f_3(t) = 1 (at that point, the fluctuations due to individual stars dominate). When d >= 5, the large deviation is due to a thin tube (a wormhole) along which the star density increases monotonically, until the point f_d(t) = 1 (where again fluctuations due to individual stars dominate). In dimension 4 we observe a double phase transition, where the transition between low- dimensional behavior (attracting galaxy) and high-dimensional behavior (wormhole) occurs at t = 4/3. As consequences, we determine the tail behavior of the distance from a star to a uniform point in its cell, and prove a sharp lower bound for the tail probability of the cell's diameter, matching our earlier upper bound.  

This is joint work with Sourav Chatterjee, Yuval Peres and Dan Romik.

Diffusion limited aggregation (DLA) in 2 or more dimensions is an infamously difficult model for the growth of a random fractal. In the model, a sequence of aggregates A_n is built on the square lattice, by starting with a single point A_0={0}, and adding one particle at each step.The position of the particle ad$ at step n is chosen by starting a simple random walk from "infinity" (far away) and letting the walk wander until it becomes a neighbour of the current aggregate A_{n-1}, at which time it is stopped and added to the aggregate to form A_n.

DLA was introduced in 1981 and attracted massive attention. (184,000 google hits). Even so, Kesten's 1987 upper bound on the diameter growth rate is almost the only proven result on it.

We define a variation of DLA in one dimension. This becomes interesting when the random walk generating the DLA has arbitrary long jumps. It turns out that the growth rate of the aggregate depends on the step distribution and more specifically on the decay of the tail opf the undrlying random walk. In particular we show that there are at least three phase transitions in the behaviour when the step distribution has finite 1/2 moment, finite variance, and finite third moment. And more suprisingly that there seems to be no first-order phase transition when the walk goes from the transient to the recurrent regimn (finite expectation).

If time permits, we will also discuss some results on the limit aggregate A_infinity, and show a transient random walk for which the aggregate eventually spans all points in Z.

Joint work with Omer Angel, Itai Benjamini and Gadi Kozma.

In the early 60s Kesten showed that self-avoiding walk in the upper half plane has a decomposition into an i.i.d. sequence of "irreducible bridges". Loosely defined, a bridge is a self-avoiding path that achieves its minimum and maximum heights at the start and end of the path (respectively), and it is irreducible if it contains no smaller bridges. Considering only the 2-dimensional case, one can ask if the (likely) scaling limit of self-avoiding walk, the SLE(8/3) process, also has such a decomposition. I will talk about recent work with Hugo Duminil from Ecole Normale Superieure that provides a positive answer, using only the restriction property of SLE(8/3). In the end we are able to decompose the SLE(8/3) path as a Poisson Point Process on the space of irreducible bridges, in a way that is similar to Ito's excursion decomposition of a Brownian motion according to its zeros. Our decomposition can actually be generalized beyond SLE(8/3) and applied to an entire family of "restriction measures", hence the title of the talk. If time permits I will also talk about the natural time parameterization for SLE(8/3), which has immediate applications towards the bridge decomposition.

Let each site of the triangular lattice, with small mesh Q$\eta$, have an independent Poisson clock with a certain rate $r(Q\eta) = \eta^{3/4+o(1)}$ switching between open and closed. Then, at any given moment, the configuration is just critical percolation; in particular, the probability of a left-right open crossing in the unit square is close to 1/2. Furthermore, because of the scaling, the expected number of switches in unit time between having a crossing or not is of unit order.

We prove that the limit (as $\eta \to 0$) of the above process exists as a Markov process, and it is conformally covariant: if we change the domain with a conformal map $\phi(z)$, then time has to be scaled locally by $|\phi'(z)|^{3/4}$. The same proof yields a similar result for near-critical percolation, and it also shows that the scaling limit of (a version of) the Minimal Spanning Tree exists, it is invariant under translations, rotations and scaling, but *probably* not under general conformal maps.

Joint work with Christophe Garban and Oded Schramm.

There is a unique "fair" allocation that is "stable" in the sense of the Gale-Shapley stable marriage problem, every point of M is assigned a a bounded region with finitely many components, but obtaining any(!) tail estimate for the diameter of these regions is open. These allocations arose from the continuum version of the "extra head" problem. The original problem is to find in a sequence of i.i.d. coins with heads probability p, one coin that landed heads so that all other coins are still i.i.d. with heads probability p [This is possible only when 1/p is an integer]. (Talk based on joint works with C. Hoffman and A. Holroyd).

November 24
Bruce Reed (McGill)
The evolution of the mixing rate

We will discuss the mixing rate of the standard random walk on the giant component of the random graph G(n,p). We tie down the mixing rate precisely for all values of p greater than (1+c)/n for any positive constant c. We need to develop a new bound on the mixing time of general Markov Chains, inspired by and extending work of Kannan and Lovasz. This is joint work with Nick Fountoulakis.

November 10
Benedek Valkó (Technical University, Budapest)
Hydrodynamic limit for perturbation of equilibria

We derive a special class of two-component systems of PDEs (hyperbolic conservation laws) as hydrodynamic limits for interacting particle systems (in the domain where the solution stays smooth). The scaling regime interpolates between the Eulerian scaling and the scaling of equilibrium fluctuations. The PDEs are derived as "universal laws" driving propagation of small perturbations of equilibria. (This is a joint work with Balint Toth.)

November 3
Nick Wormald (Waterloo)
The size of the 2-core in a random graph

Erdos and Renyi first considered the evolution of a random graph, in which n vertices begin life as isolated points and then edges are thrown in randomly one by one. This evolving random graph undergoes a phase transition when the number of edges is around n/2: a "giant" component suddenly appears.

We give a result on the joint distribution of three parameters of the giant component in the phase after it appears: the number of vertices in the 2-core (the largest subgraph of minimum degree 2 or more); the excess (#edges - #vertices) of the 2-core; and the number of vertices not in the 2-core. This uses a combination of combinatorial and probabilistic tools. It is joint work with B. Pittel.

October 27
David Revelle (UC Berkeley)
Mixing times for random walks on finite lamplighter groups

We study random walks on lamplighter groups. In a wide class of examples, we examine how different notions of of mixing time are related to maximal hitting time, expected cover time and the relaxation time of the underlying graph.

For the case of the a lamplighter group over the torus, the relaxation time is of the order n² log n, the total variation mixing time is on the order of n² log² n, and the uniform mixing time is on the order of n⁴.
This is joint work with Y. Peres.

September 29
Shlomo Hoory (UofT computer science)
An Alon-Boppana type bound for irregular graphs

Consider a finite connected graph with average degree d>=2. Using random walks, we will prove that the spectral radius of its universal cover must be at least 2 sqrt(d-1).
Using the above we generalize the Alon-Boppana theorem to irregular graphs. Assume that a graph has an r-robust average degree, i.e. the average degree after deleting any radius r ball is at least d. We give a lower bound on the second largest eigenvalue of the adjacency matrix in terms of r and d.

September 22
Ben Morris (Bloomington)
The mixing time for simple exclusion

We obtain a tight bound of O(L² log r) for the mixing time of the exclusion process in Z^d/LZ^d with r <= L^d/2 particles.