Cutoffs for exclusion and interchange processes on finite graphs
Abstract
We prove a general theorem on cutoffs for symmetric exclusion and interchange processes on finite graphs $G_N=(V_N,E_N)$, under the assumption that either the graphs converge geometrically and spectrally to a compact metric measure space, or they are isomorphic to discrete Boolean hypercubes. Specifically, cutoffs occur at times $\displaystyle t_N= (2\gamma_1^N)^{1}\log V_N$, where $\gamma_1^N$ is the spectral gap of the symmetric random walk process on $G_N$. Under the former assumption, our theorem is applicable to the said processes on graphs such as: the $d$dimensional discrete grids and tori for any integer dimension $d$; the $L$th powers of cycles for fixed $L$, a.k.a. the $L$adjacent transposition shuffle; and selfsimilar fractal graphs and products thereof.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 arXiv:
 arXiv:2010.16227
 Bibcode:
 2020arXiv201016227C
 Keywords:

 Mathematics  Probability;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Combinatorics;
 35K05;
 82C22;
 60B10;
 60B15;
 60J27
 EPrint:
 There is a gap in the proof in Section 5 of the paper