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 $S_{\star }\subset S$. 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.

34 pages

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• Derivation of mean-field equations for stochastic particle systems
• Monotonicity and Condensation in Homogeneous Stochastic Particle Systems
• Metastability of Ising and Potts models without external fields in large volumes at low temperatures
• Metastable Markov chains
• Metastable behavior of weakly mixing Markov chains: the case of reversible, critical zero-range processes
• Size-biased diffusion limits and the inclusion process
• Second Time Scale of the Metastability of Reversible Inclusion Processes
• Spectral gap of the symmetric inclusion process
• Full $\Gamma$-expansion of reversible Markov chains level two large deviations rate functionals
• Condensation in zero-range processes with a fast rate
• Generalized Dirichlet and Thomson Principles and Their Applications