Metastability in the reversible inclusion process

Alessandra Bianchi,Cristian Giardinà,Sander Dommers

doi:10.1214/17-ejp98

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 $S_\star\subset S$. 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.

Highlights

The inclusion process is an interacting particle system introduced in the context of non-equilibrium statistical mechanics, as a dual process of certain diffusion processes modeling heat conduction and Fourier’s law [18, 19, 20]
It is related to models in mathematical population genetics [13], such as the Moran model, and to models of wealth distribution [15]
The inclusion process is interesting in its own right as an interacting particle system belonging to the class of misanthrope processes [16]

Summary

Introduction

The inclusion process is an interacting particle system introduced in the context of non-equilibrium statistical mechanics, as a dual process of certain diffusion processes modeling heat conduction and Fourier’s law [18, 19, 20]. Particles jump over a set S of vertices, the total number of particles N is conserved by the dynamics. The transitions are driven by two competing contributions to the total jump rate. Denoting by ηx the particle number at site x, and calling r : S × S → R+ the jump rates of a continuous-time irreducible random walk on S, the process is defined by the following rules (see Section 2.1 for the process generator): ii) secondly, particles jump because of an attractive interaction: each of the ηx particles at site x waits a random time which is the minimum of exponential clocks of parameters ηzr(x, z) for z ∈ S, and jumps to site y with probability ηyr(x, y)/( z∈S ηzr(x, z))

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Conclusion