## Electronic Journal of Probability

### Metastability in the reversible inclusion process

#### Abstract

We study the condensation regime of the finite reversible inclusion process, i.e., the inclusion process on a finite graph $S$ with an underlying random walk that admits a reversible measure. We assume that the random walk kernel is irreducible and its reversible measure takes maximum value on a subset of vertices $S_{\star }\subseteq S$. We consider initial conditions corresponding to a single condensate that is localized on one of those vertices and study the metastable (or tunneling) dynamics. We find that, if the random walk restricted to $S_{\star }$ is irreducible, then there exists a single time-scale for the condensate motion. In this case we compute this typical time-scale and characterize the law of the (properly rescaled) limiting process. If the restriction of the random walk to $S_{\star }$ has several connected components, a metastability scenario with multiple time-scales emerges. We prove such a scenario, involving two additional time-scales, in a one-dimensional setting with two metastable states and nearest-neighbor jumps.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 70, 34 pp.

Dates
Accepted: 21 August 2017
First available in Project Euclid: 13 September 2017

https://projecteuclid.org/euclid.ejp/1505268101

Digital Object Identifier
doi:10.1214/17-EJP98

Mathematical Reviews number (MathSciNet)
MR3698739

Zentralblatt MATH identifier
06797880

#### Citation

Bianchi, Alessandra; Dommers, Sander; Giardinà, Cristian. Metastability in the reversible inclusion process. Electron. J. Probab. 22 (2017), paper no. 70, 34 pp. doi:10.1214/17-EJP98. https://projecteuclid.org/euclid.ejp/1505268101

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