The cutoff profile for the simple exclusion process on the circle

Abstract

In this paper, we give a very accurate description of the way the simple exclusion process relaxes to equilibrium. Let $P_t$ denote the semi-group associated the exclusion on the circle with $2N$ sites and $N$ particles. For any initial condition $\chi$, and for any $t\ge \frac{4N^2}{9{\pi }^2}\log N$, we show that the probability density $P_t(\chi ,\cdot )$ is given by an exponential tilt of the equilibrium measure by the main eigenfunction of the particle system. As $\frac{4N^2}{9{\pi }^2}\log N$ is smaller than the mixing time which is $\frac{N^2}{2{\pi }^2}\log N$, this allows to give a sharp description of the cutoff profile: if $d_N(t)$ denote the total-variation distance starting from the worse initial condition we have \[\lim_{N\to\infty}d_N\biggl(\frac{N^2}{2π^2}\log N+\frac{N^2}{π^2}s\biggr)=\operatorname {erf}\biggl(\frac{\sqrt{2}}πe^{-s}\biggr),\] where $\mathrm{erf}$ is the Gauss error function.

Published at http://dx.doi.org/10.1214/15-AOP1053 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• The mixing time for simple exclusion

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