Electronic Journal of Probability

Metastability of hard-core dynamics on bipartite graphs

Frank den Hollander, Francesca R. Nardi, and Siamak Taati

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We study the metastable behaviour of a stochastic system of particles with hard-core interactions in a high-density regime. Particles sit on the vertices of a bipartite graph. New particles appear subject to a neighbourhood exclusion constraint, while existing particles disappear, all according to independent Poisson clocks. We consider the regime in which the appearance rates are much larger than the disappearance rates, and there is a slight imbalance between the appearance rates on the two parts of the graph. Starting from the configuration in which the weak part is covered with particles, the system takes a long time before it reaches the configuration in which the strong part is covered with particles. We obtain a sharp asymptotic estimate for the expected transition time, show that the transition time is asymptotically exponentially distributed, and identify the size and shape of the critical droplet representing the bottleneck for the crossover. For various types of bipartite graphs the computations are made explicit. Proofs rely on potential theory for reversible Markov chains, and on isoperimetric results.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 97, 65 pp.

Received: 12 January 2018
Accepted: 8 August 2018
First available in Project Euclid: 21 September 2018

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 82C27: Dynamic critical phenomena

interacting particle systems bipartite graphs potential theory metastability isoperimetric problems

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den Hollander, Frank; Nardi, Francesca R.; Taati, Siamak. Metastability of hard-core dynamics on bipartite graphs. Electron. J. Probab. 23 (2018), paper no. 97, 65 pp. doi:10.1214/18-EJP210. https://projecteuclid.org/euclid.ejp/1537495434

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