Condensation and Metastable Behavior of Non-reversible Inclusion Processes

TL;DR

This paper extends the analysis of metastable behavior in inclusion processes to non-reversible cases, overcoming the challenge of unknown invariant measures and revealing new differences in time scales and limiting processes.

Contribution

It provides the first rigorous quantitative analysis of metastability in non-reversible inclusion processes without explicit invariant measures.

Findings

Time scale and limiting process differ from reversible cases

Identifies three distinct time scales in large torus limit

First rigorous results on metastability without explicit invariant measure

Abstract

In this article, we perform quantitative analyses of metastable behavior of an interacting particle system known as the inclusion process. For inclusion processes, it is widely believed that the system nucleates the condensation of particles because of the attractive nature of the interaction mechanism. The metastable behavior of the inclusion processes corresponds to the movement of the condensate on a suitable time scale, and the computation of the corresponding time scale and the characterization of the scaling limit of the condensate motion are the main problems in the study of metastability of inclusion processes. Previously, these problems were solved for reversible inclusion processes in [Bianchi, Dommers, and Giardin\`a, Electronic Journal of Probability, 22: 1-34, 2017], and the main contribution of the present study is to extend this analysis to a wide class of non-reversible inclusion processes. Non-reversibility is a major obstacle to analyzing such models, mainly because there is no closed-form expression of the invariant measure for the general case, and our main achievement is to overcome this difficulty. In particular, our results demonstrate that the time scale and limiting process of non-reversible inclusion processes are quantitatively and qualitatively different from those of reversible ones, respectively. We emphasize that, to the best of our knowledge, these results are the first rigorous quantitative results in the study of metastability when the invariant measure is not explicitly known. In addition, we consider the thermodynamic limit of metastable behavior of inclusion processes on large torus as in the paper [Armend\'ariz, Grosskinsky, and Loulakis, Probability Theory and Related Fields, 169: 105-175, 2017]. For this model, we observe three different time scales according to the level of asymmetry of the model.