The cutoff profile for exclusion processes in any dimension
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 $t_N=\logV_N/(2\lambda^N_1)$, 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 selfsimilar 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 infinitedimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 DOI:
 10.48550/arXiv.2106.03685
 arXiv:
 arXiv:2106.03685
 Bibcode:
 2021arXiv210603685C
 Keywords:

 Mathematics  Probability;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics;
 60J27;
 82C22;
 60B10;
 35K05
 EPrint:
 39 pages, 6 figures. This paper replaces the retracted preprints arXiv:2010.16227 and arXiv:2011.08718, along with a change of authors. The cutoff problem for the interchange process is not addressed