New Directions in Probability Theory

Session:  Random Media
Organizer:  Michael Cranston, Univ. of California, Irvine and Univ. of Rochester

Abstract:

We shall discuss several asymptotic problems for transport by random flows. In particular, in our work with D.Dolgopyat and V.Kaloshin we describe the long-time behavior of connected sets carried by the flow. The analysis relies on the estimates of the mixing rate of the multi-point motion.

Abstract:

We give a detailed description of the transition between the annealed and quenched descriptions of the dynamics of branching Random Walks in random environment. This transition will be given as a consequence of the nature of the edge of the spectum for the Parabolic Anderson problem in a large volume. This is joint work with A.Ramirez (Santiago) and S.Molchanov (Charlotte).

Abstract:

The Anderson model was originally conceived to decide the question of entrapment of electrons in crystals with impurities. This situation was modelled using a Hamiltonian which consisted of a discrete Laplacian (on $\mathbf Z^d$) plus a random potential indexed by sites in $\mathbf Z^d. $ Similar operators appear in a wide variety of applications from the generation of stellar magnetic fields to the population dynamics of plankton. In this talk we will consider solutions of equations of the form $$ \partial u(t,x)/\partial t = \kappa \Delta u(t,x) + \zeta (t,x) u(t,x),\,u(0,x)=u_0(x), $$ where $\Delta$ is the Laplacian on the appropriate space and $\zeta (t,x)$ is a family of random variables. We will show existence of the limit $$\lim_{t \rightarrow \infty }\frac{1}{t}\log u(t,x)=\lambda(\kappa)$$ with probability one in a variety of contexts. The number $\lambda(\kappa)$ is called the almost sure Lyapunov exponent. We will also explain the asymptotic behavior of $\lambda(\kappa)$ as $\kappa \searrow 0. $ The talk is based on three recent joint papers with Mountford and Shiga and Mountford.

Session:  Random Matrices
Organizer:  Craig Tracy, University of California, Davis

Abstract:

A review of some recent results on computing the partition function, correlation functions and spacing distributions for two-matrix models, both of finite size and in the N -> infinity will be given. Emphasis will be placed on the role of "duality", the isomonodromic deformation systems characterizing biorthogonal polynomials and the Riemann-Hilbert problems governing both large x (and y) asymptotics and spacing distributions for the eigenvalues. Recent results concerning large N limits and their relation to dispersionless integrable flows will also be discussed. (Based on joint work with M. Bertola and B, Eynard.)

Abstract:

I will consider, from the perspective of free probability, some basic questions about random matrices. In particular, I will address multi-matrix models and (global) fluctuations of eigenvalues.

Session:  Superprocesses
Organizer:  Tom Salisbury, York University

Abstract:

We will discuss the ergodic behaviour of systems of particles performing independent random walks, binary splitting, coalescence and deaths. This is joint work with Jan Swart.

Abstract:

This talk will deal with the heat equation with a random potential given by time-independent Levy noise. To avoid singularities, we set up the equation using a modified version of the Wick product. We give criteria for convergence of our expansions which define the solution. 

Abstract:

Duality is a powerful tool in the study of interacting stochastic systems. There is a natural dual relationship between two systems of coalescing Brownian motions. Loosely put, the joint distribution of a system of coalescing Brownian motions is determined by an evolving partition of the real line driven by another system of coalescing Brownian motions. Such a duality can be used to study continuous-site stepping-stone models. It also allows some explicit calculations on a superprocess with coalescing Brownian spatial motion. 

Session:  Self-Avoiding Walk
Organizer:  Greg Lawler, Cornell University

Abstract:

Self-avoiding walk in four dimensions is conjectured to have an end-to-end distance that grows as $N^{1/2}\log^{1/8}N$ and a Green's function that behaves, for the critical killing rate, as $|x-y|^{-2}$. I will describe results in this direction which have been obtained for a weakly self-avoiding walk on a four dimensional hierarchical lattice.

Abstract:

The conjectured conformal invariance of the scaling limit of the two-dimensional self-avoiding walk has led to explicit predictions about the scaling limit (Lawler, Schramm and Werner) that can be tested by Monte Carlo simulations. We can also test the conformal invariance by comparing the simulations of the walk in two different simply connected domains. We will present some of these simulations as well as some simulations testing the possible conformal invariance of some other self-avoiding and weakly self-avoiding walks.

Abstract:

Knots in long polymer molecules have been studied for many years, and have particular relevance to certain aspects of DNA behaviour. A simple model for polymers is the self-avoiding walk, so one is led naturally to consider questions about knottedness of random (closed) self-avoiding walks. We will present a mathematical survey what is known and what is not known on this topic.

Session:  Markov Chains and Algorithms
Organizer:  Robin Pemantle, University of Pennsylvania

Abstract:

For a given graph G and integer C which is greater than the maximum degree of G, we consider generating a random C-colouring of G as follows. We start with an arbitrary C-colouring, and at each step we choose a uniformly random vertex v, and a uniformly random colour c from the set of colours which do not appear on any neighbours of v. Then we change the colour of v to c. This commonly studied procedure is a Markov chain known as the Glauber dynamics. The question usually asked is whether the procedure is rapidly mixing, i.e. whether the colouring generated after some polynomial number of steps is "close" to being uniformly random. It is conjectured that on any graph, the procedure is indeed rapidly mixing so long as the number of colours is at least D + 2, where D is the maximum degree The best result in this direction (due to Vigoda) is that this is true so long as the number of colours is at least (11/6)D.In this talk, we will survey a sequence of results that substantially reduce the lower bound of (11/6)D on the colours so long as the graph has sufficiently high girth and maximum degree.

Abstract:

The Coupling Method is a classical technique for proving rapid convergence of a Markov chain. It is noteworthy both for its theoretical power (there always exists a coupling with optimal convergence rate), and for its practicality: it is easy to use, and tends to produce clear and insightful proofs. We will present some recent extensions to the Coupling Method, which attempt to realize more of its theoretical power, without sacrificing too much clarity or ease of use. 

Abstract:

The maximum displacement of a branching random walk is well understood, but how long a search is required to find a nearly optimal branch? We study this in the simplest case, where a binary tree is labeled with IID Bernoulli(p) variables with $p \leq 1/2$. Let $M_n$ denote the maximum number of ones in a rooted path of length n. It is known that $$ M_n = c(p) n - g(n) + O(1) $$where $g(n) = c'(p) \log n$ for the subcritical case, $p < 1/2$, and $g(n) = c'' \log \log n$ at criticality ($p = 1/2$). We consider here the complexity of finding a path whose sum is at least $(1-r) M_n$. We show this is of order $r^{-1} n$ in the critical case, and at least $C r^{-3/2} n / \log2 r$ in the subcritical case; we are working on the upper bound in the subcritical case. (Joint work with Yuval Peres, U.C. Berkeley)

Invited Lectures

Abstract:

A self-avoiding walk (SAW) in the lattice is a nearest neighbor walk conditioned to have no self-intersections. The problem of the self-avoiding walk is still open in the hardest dimensions, two, three, and four. After reviewing the SAW problem in all dimensions, I will describe what the continuum limit of SAW in two dimensions is conjectured to be. It turns out that the assumption of a conformally invariant allows us to determine the limit. This is joint work with Oded Schramm and Wendelin Werner.

Abstract:

Tracy_Abstract.pdf

Medallion Lectures

Abstract:

Measures from non-intersecting paths naturally give rise to determinantal processes. We will consider some examples related to random growth and random tilings. Scaling limits of these models lead to the so called Airy process. Determinantal processes also occur naturally in random matrix theory, and the Airy process can also be obtained from Dyson's Brownian motion on Hermitian matrices.

Abstract:

A century ago Einstein postulated that the origin of the Brownian motion was due to the light water molecules continuously bombarding the heavy pollen. This explained the Brownian motion in the framework of the Newtonian mechanics. Since the discovery of quantum mechanics it has been a major challenge to verify the emergence of diffusion from the Schrödinger equation. In this talk I will report a rigorous derivation of a diffusion equation from a scaling limit of a random Schrödinger equation in a weak random potential. This is a joint work with L. Erdos and M. Salmhofer.