Spatial shuffling: Large Cycles in the Interchange ​Process in dimension 5

The interchange process $\sigma_T$ is a permutation valued stochastic process on a graph evolving in time by transpositions on its edges at rate 1. The cycles generated can be viewed as part of a larger class of self-interacting random walks. 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.

Joint work with Dor Elboim.

You can learn more about Professor Allan Sly here: https://web.math.princeton.edu/~asly/