Cutoff for the Glauber-Exclusion process in the full high-temperature regime: an information percolation approach

TL;DR

This paper establishes cutoff phenomena for the Glauber-Exclusion process in high-temperature regimes using an information percolation approach, linking hydrodynamic limits to mixing times in low-dimensional lattice systems.

Contribution

It introduces a novel application of the information percolation framework to analyze cutoff in the Glauber-Exclusion process, including explicit invariant measure and spectral gap results.

Findings

Cutoff occurs in the high-temperature regime for dimensions 1 and 2.

Explicit formula for the invariant measure is derived.

Pre-cutoff is established in all dimensions.

Abstract

The Glauber-Exclusion process is a superposition of a Glauber dynamics and the Symmetric Simple Exclusion Process (SSEP) on the lattice. The model was shown to admit a reaction-diffusion equation as the hydrodynamic limit. In this article, we define a notion of temperature regimes via the reaction function in the equation and prove cutoff in the full high-temperature regime for the attractive model in dimensions $1$ and $2$ with periodic boundary condition. Our results show that the equation in the hydrodynamic limit reflects the mixing behavior of the large but finite system. Besides, cutoff is proved under the lack of reversibility and an explicit formula for the invariant measure. We also provide the spectral gap and prove pre-cutoff in all dimensions. Our proof involves a new interpretation of attractiveness, the information percolation framework introduced by Lubetzky and Sly, anti-concentration of simple random walk on the lattice, and a coupling inspired by excursion theory. We hope that this approach can find new applications in the future.