Electronic Journal of Probability

Mixing times for exclusion processes on hypergraphs

Stephen B. Connor and Richard J. Pymar

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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 ([14]) and developed by Oliveira ([15]) for studying the exclusion process on a graph.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 73, 48 pp.

Received: 23 March 2018
Accepted: 9 June 2019
First available in Project Euclid: 28 June 2019

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Digital Object Identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35]

mixing time exclusion interchange random walk hypergraph coupling

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Connor, Stephen B.; Pymar, Richard J. Mixing times for exclusion processes on hypergraphs. Electron. J. Probab. 24 (2019), paper no. 73, 48 pp. doi:10.1214/19-EJP332. https://projecteuclid.org/euclid.ejp/1561687606

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