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 self-similar fractal graphs and products thereof.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2020
- arXiv:
- arXiv:2010.16227
- Bibcode:
- 2020arXiv201016227C
- Keywords:
-
- Mathematics - Probability;
- Condensed Matter - Statistical Mechanics;
- Mathematics - Combinatorics;
- 35K05;
- 82C22;
- 60B10;
- 60B15;
- 60J27
- E-Print:
- There is a gap in the proof in Section 5 of the paper