Metastability in the reversible inclusion process

TL;DR

This paper investigates the metastable behavior of the reversible inclusion process on finite graphs, identifying single and multiple time-scale dynamics depending on the structure of the underlying random walk.

Contribution

It characterizes the metastable dynamics and time-scales of the inclusion process, including cases with multiple metastable states and complex tunneling behavior.

Findings

Single time-scale for condensate motion when the restricted walk is irreducible.

Explicit computation of the typical metastable time-scale.

Multiple time-scales scenario in one-dimensional models with several metastable states.

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.