SCIENTIFIC PROGRAMS AND ACTIVITIES

Toronto Probability Seminar 2010-11
held at the Fields Institute

Organizers
Bálint Virág , Benedek Valkó
University of Toronto, Mathematics and Statistics

For questions, scheduling, or to be added to the mailing list, contact the organizers at:
probsem-at-math-dot-toronto-dot-edu

Speaker and Talk Title

Fredrik Viklund (Columbia)
On Convergence Rates to SLE for Random Lattice Curves

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.

Mark Holmes (Auckland)
Random walks in degenerate random environments

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).

Mladen Savov (Oxford)
Exponential Functional of Lévy Processes

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)).

Maurice Duits (Caltech)
An equilibrium problem for the two matrix model with a quartic potential

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.

Enza Orlandi (Universita' di Roma Tre)
Ginzburg Landau functional with external random field: minimizers and interfaces

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 Gonalves (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 graphs 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.

*SPECIAL SEMINAR
Mar. 1, 2011
N638 Ross Building, York University
10:30 a.m.

*SPECIAL SEMINAR
Lerna Pehlivan (Carleton University)
Top to random shuffles and number of fixed points

The number of fixed points is studied for random permutations and for some shuffles such as riffle shuffles. We look at the same problem for top to random shuffles. We provide the formulas for the expected value and the variance of the number of fixed points of a permutation obtained after a number of top to random shuffles.

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.

Vladislav Vysotsky
Positivity of Integrated Random Walks

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.

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.

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

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.

Dec.6, 2010
Stewart Library
4:00pm

PLEASE NOTE DATE AND TIME
Nov. 26, 2010
Stewart Library
2:40pm

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

Leonid Pastur (National Academy of Sciences of Ukraine/Fields)
Limiting Fluctuation Laws for Spectral Statistics of Random Matrices

Misha Sodin (Tel Aviv University)
Random complex zeros: fluctuations and correlations

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.

Daniel Remenik (University of Toronto)
Brunet-Derrida particle systems, free boundary problems, and Wiener-Hopf equations

We consider a branching-selection system in R 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 $c<a$, where $a$ is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of Wiener-Hopf equations.

Joint work with Rick Durrett.

Alan Hammond (Oxford University)
Biased motion in disorder: persistent discreteness, rational resonance, and stable limits

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).

Tom Alberts (University of Toronto)
Intermediate Disorder for Directed Polymers in Dimension 1+1, and the Continuum Random Polymer

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 shows 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.

Arnab Sen (UC Berkeley)
Coalescing systems of non-Brownian particles

A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the starting set is compact. We investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous. We show that Arratia's conclusion is valid for Brownian motions on the Sierpinski gasket and for stable processes on the real line with stable index greater than one.

Joint work with Steve Evans and Ben Morris.

Ivan Corwin (Courant-NYU)
Fluctuations for the KPZ Universality Class

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.