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September 2016 The cutoff profile for the simple exclusion process on the circle
Hubert Lacoin
Ann. Probab. 44(5): 3399-3430 (September 2016). DOI: 10.1214/15-AOP1053


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

\[\lim_{N\to\infty}d_{N}(\frac{N^{2}}{2\pi^{2}}\log N+\frac{N^{2}}{\pi^{2}}s)=\operatorname{erf}(\frac{\sqrt{2}}{\pi}e^{-s}),\] where $\operatorname{erf}$ is the Gauss error function.


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Hubert Lacoin. "The cutoff profile for the simple exclusion process on the circle." Ann. Probab. 44 (5) 3399 - 3430, September 2016.


Received: 1 March 2015; Revised: 1 July 2015; Published: September 2016
First available in Project Euclid: 21 September 2016

zbMATH: 06653521
MathSciNet: MR3551201
Digital Object Identifier: 10.1214/15-AOP1053

Primary: 37L60 , 60J10 , 82C20

Keywords: cutoff profile , Markov chains , mixing time , Particle systems

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 5 • September 2016
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