December  7, 2023

May 15-19, 2006

Workshop on Random Walks in Random Environments
The Fields Institute




Christina Goldschmidt, Cambridge:
Coagulation-fragmentation duality, Poisson-Dirichlet distributions and random recursive trees

We give a new example of duality between certain coagulation and fragmentation operators. More specifically, if we start with a random variable with Poisson-Dirichlet PD(alpha, theta) distribution then, after application of our fragmentation operator, we obtain a random variable with PD(alpha, theta + 1) distribution. The coagulation operator goes back the other way. These relations provide a counterpart to Pitman's relations between PD(alpha, theta) and PD(alpha x beta, theta). Repeated application of our fragmentation operator gives rise to a Markov fragmentation chain, which can be encoded naturally by certain random recursive trees. [Based on joint work with Rui Dong (Berkeley) and James Martin (Oxford).]

Russell Lyons, Indiana:
Unimodularity and Stochastic Processes

Stochastic processes on vertex-transitive graphs, especially Cayley graphs of groups, have been studied for 50 years (not counting the special case of integer lattices, which goes back hundreds of years). The assumption of invariance under graph automorphisms plays a key role, but investigations of the last 15 years have shown that an additional assumption is also extremely useful. This newer assumption is the property of unimodularity, which is equivalen to the Mass-Transport Principle. We shall review some well-known applications and also discuss recent work with David Aldous. This includes three theorems on RWRE.

Robin Pemantle, Dept. of Math, Univ. of Pennsylvania:
Multivariate generating function techniques and an application to quantum random walks

A number of problems in combinatorics and probability may be encoded into a generating function, F, and limit theorems extracted analytically. The extraction depends on details of the function F. I will discuss the case where F = G/H is a rational 3-variable function
and H(1+x,1+y,1+z) is a conic.

Why should you care about this case? It turns out that this encompasses many cases of the "Arctic Circle" phenomena:i in tiling examples, randomness is confined to a linearly growing disk; the location of a quantum random walk is uniformly spread over such a disk.
This work is in progress and is joint with Yuliy Baryshnikov.

Yuval Peres, Berkeley:
The Unreasonable Effectiveness of Martingales

Martingales are often used by combinatorialists for their concentration properties. The goal of this survey talk is to illustrate the usefullness of optional stopping arguments for natural problems on critical random graphs, random walks, and embeddings of graphs in Euclidean space. In particular the same martingale lemma can be used to prove
(1) the two-thirds power law for the largest component of the
critical random graph G(n,1/n),
(2) the same law for critical percolation on a random 3-regular graph, and
(3) the -1/2 power law bound for rate of decay of the transition
probabilities p^k(x,y) of random walk on any graph of bounded degree. (Talk based on joint works with Asaf Nachmias and with Ben Morris)

For further information please contact gensci(PUT_AT_SIGN_HERE)