Cutoffs for exclusion processes on graphs with open boundaries

TL;DR

This paper establishes cutoff phenomena for symmetric exclusion processes on various graphs with open boundaries, including grids and fractals, by analyzing a rescaled density fluctuation field.

Contribution

It provides a general theorem on cutoff for these processes under geometric and spectral convergence assumptions, extending to complex graph structures.

Findings

Cutoff occurs in symmetric exclusion processes on diverse graphs.

The cutoff time matches upper and lower bounds derived through different methods.

The approach applies to fractal and product graphs, broadening previous results.

Abstract

We prove a general theorem on cutoffs for symmetric simple exclusion processes on graphs with open boundaries, under the natural assumption that the graphs converge geometrically and spectrally to a compact metric measure space with Dirichlet boundary condition. Our theorem is valid on a variety of settings including, but not limited to: the $d$-dimensional grid for every integer dimension $d$; and self-similar fractal graphs and products thereof. Our method of proof is to identify a rescaled version of the density fluctuation field---the cutoff martingale---which allows us to prove the mixing time upper bound that matches the lower bound obtained via Wilson's method.