The cutoff profile for exclusion processes in any dimension

TL;DR

This paper establishes the precise cutoff profile for symmetric exclusion processes on various graphs converging to metric spaces, identifying the cutoff time and describing the limiting total variation profile using an analytic approach.

Contribution

It proves the cutoff phenomenon and its profile for exclusion processes on general spaces, introducing cutoff semimartingales and employing the entropy method for analysis.

Findings

Cutoff occurs at time t_N = log|V_N| / (2λ_1^N).

The total variation profile converges to a limit described by Brownian motion.

Results hold on Euclidean lattices and fractal spaces.

Abstract

Consider symmetric simple exclusion processes, with or without Glauber dynamics on the boundary set, on a sequence of connected unweighted graphs $G_N=(V_N,E_N)$ which converge geometrically and spectrally to a compact connected metric measure space. Under minimal assumptions, we prove not only that total variation cutoff occurs at times $t_N=\log|V_N|/(2\lambda^N_1)$, where $|V_N|$ is the cardinality of $V_N$, and $\lambda^N_1$ is the lowest nonzero eigenvalue of the nonnegative graph Laplacian; but also the limit profile for the total variation distance to stationarity. The assumptions are shown to hold on the $D$-dimensional Euclidean lattices for any $D\geq 1$, as well as on self-similar fractal spaces. Our approach is decidedly analytic and does not use extensive coupling arguments. We identify a new observable in the exclusion process -- the cutoff semimartingales -- obtained by scaling and shifting the density fluctuation fields. Using the entropy method, we prove a functional CLT for the cutoff semimartingales converging to an infinite-dimensional Brownian motion, provided that the process is started from a deterministic configuration or from stationarity. This reduces the original problem to computing the total variation distance between the two versions of Brownian motions, which share the same covariance and whose initial conditions differ only in the coordinates corresponding to the first eigenprojection.