Large cycles for the interchange process

Abstract: The interchange process $\sigma_T$ is a random permutation valued stochastic process on a graph evolving in time by transpositions on its edges at rate 1. On $Z^d$, when $T$ is small all the cycles of the permutation $\sigma_T$ are finite almost surely but it is conjectured that infinite cycles appear in dimensions 3 and higher for large times.  In this talk I will focus on the finite volume case where we establish that macroscopic cycles with Poisson-Dirichlet statistics appear for large times in dimensions 5 and above.

Bio: Allan Sly is the Anthony H. P. Lee ’79 Professor of Mathematics at Princeton University. His research is in discrete probability theory and its applications to problems from statistical physics, theoretical computer science and theoretical statistics. Most of his work is centered on stochastic processes on networks in a range of different settings. Two major focuses are the analysis of the mixing times of Markov chains, particularly Glauber dynamics and the role phase transitions play in the computational complexity and in probabilistic models more generally. He completed his PhD in Statistics at UC Berkeley in 2009 and has been a postdoc in the Theory Group at Microsoft Research.