Speaker: Dor Elboim (Princeton University) 

Title: Infinite cycles in the interchange process in five dimensions.

Abstract:

In the interchange process on a graph G=(V,E), there is a particle on each vertex of the graph and an independent Poisson clock on each one of the edges. Once the clock of an edge rings, the two particles on the two sides of the edge switch. After time t, the particles are permuted according to a random permutation $\pi_t:V\to V$. A famous conjecture of Balint Toth states that the following holds when $G=\mathbb$ $Z^d$ :
 

(1) If d=2, then the permutation $\pi_t$ contains only finite cycles for all t>0.
(2) If $d\ge 3$, then there exists $t_c$ such that for $tt_c$, $\pi_t$ contains infinite cycles.

We prove the existence of infinite cycles for all $d\ge 5$ and all $t$ sufficiently large. To this end, we study the cyclic time random walk obtained by exposing the cycle of the origin in $\pi_t$. We show that this walk is diffusive using a multiscale inductive argument.

This is a joint work with Allan Sly.