Mixing times for exclusion processes on hypergraphs

TL;DR

This paper extends the exclusion process to hypergraphs, providing an upper bound on mixing times that relates to the two-particle mixing time, and demonstrates the bound's optimality through specific hypergraph examples.

Contribution

It introduces a hypergraph extension of the exclusion process and establishes an optimal upper bound on its mixing time based on two-particle mixing times.

Findings

Upper bound on mixing time for hypergraph exclusion process

Existence of hypergraphs where one-particle and two-particle mixing times differ significantly

Adaptation of the chameleon process for hypergraph analysis

Abstract

We introduce a natural extension of the exclusion process to hypergraphs and prove an upper bound for its mixing time. In particular we show the existence of a constant $C$ such that for any connected, regular hypergraph $G$ within some natural class, the $\varepsilon$-mixing time of the exclusion process on $G$ with any feasible number of particles can be upper-bounded by $CT_{\text{EX}(2,G)}\log(|V|/\varepsilon)$, where $|V|$ is the number of vertices in $G$ and $T_{\text{EX}(2,G)}$ is the 1/4-mixing time of the corresponding exclusion process with just two particles. Moreover we show this is optimal in the sense that there exist hypergraphs in the same class for which $T_{\mathrm{EX}(2,G)}$ and the mixing time of just one particle are not comparable. The proofs involve an adaptation of the chameleon process, a technical tool invented by Morris and developed by Oliveira for studying the exclusion process on a graph.