Universality of cutoff for exclusion with reservoirs

TL;DR

This paper analyzes the cutoff phenomenon in reversible exclusion processes with reservoirs on arbitrary networks, establishing conditions for cutoff and providing explicit examples across various dimensions and norms.

Contribution

It introduces a general characterization of spectral gap and cutoff conditions for exclusion processes on arbitrary networks, extending previous results to higher dimensions and multiple norms.

Findings

Cutoff occurs if and only if the product condition is satisfied.

Explicit cutoff examples on multidimensional lattices with various boundary conditions.

Cutoff phenomena observed in multiple norms including entropy and separation distance.

Abstract

We consider the reversible exclusion process with reservoirs on arbitrary networks. We characterize the spectral gap, mixing time, and mixing window of the process, in terms of certain simple statistics of the underlying network. Among other consequences, we establish a non-conservative analogue of Aldous's spectral gap conjecture, and we show that cutoff occurs if and only if the product condition is satisfied. We illustrate this by providing explicit cutoffs on discrete lattices of arbitrary dimensions and boundary conditions, which substantially generalize recent one-dimensional results. We also obtain cutoff phenomena in relative entropy, Hilbert norm, separation distance and supremum norm. Our proof exploits negative dependence in a novel, simple way to reduce the understanding of the whole process to that of single-site marginals. We believe that this approach will find other applications.