Cutoffs for exclusion and interchange processes on finite graphs

TL;DR

This paper establishes cutoff phenomena for symmetric exclusion and interchange processes on finite graphs, showing that mixing times are closely linked to spectral gaps and graph convergence properties.

Contribution

It provides a general theorem on cutoff times for these processes under geometric and spectral convergence assumptions, applicable to various graph families.

Findings

Cutoff occurs at times proportional to log of the number of vertices.

Mixing times depend on the spectral gap of the underlying random walk.

Results apply to grids, tori, hypercubes, and fractal graphs.

Abstract

We prove a general theorem on cutoffs for symmetric exclusion and interchange processes on finite graphs $G_N=(V_N,E_N)$, under the assumption that either the graphs converge geometrically and spectrally to a compact metric measure space, or they are isomorphic to discrete Boolean hypercubes. Specifically, cutoffs occur at times $\displaystyle t_N= (2\gamma_1^N)^{-1}\log |V_N|$, where $\gamma_1^N$ is the spectral gap of the symmetric random walk process on $G_N$. Under the former assumption, our theorem is applicable to the said processes on graphs such as: the $d$-dimensional discrete grids and tori for any integer dimension $d$; the $L$-th powers of cycles for fixed $L$, a.k.a. the $L$-adjacent transposition shuffle; and self-similar fractal graphs and products thereof.