Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The simple exclusion process on the circle has a diffusive cutoff window

Hubert Lacoin

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Abstract

In this paper, we investigate the mixing time of the simple exclusion process on the circle with $N$ sites, with a number of particle $k(N)$ tending to infinity, both from the worst initial condition and from a typical initial condition. We show that the worst-case mixing time is asymptotically equivalent to $(8\pi^{2})^{-1}N^{2}\log k$, while the cutoff window is identified to be $N^{2}$. Starting from a typical condition, we show that there is no cutoff and that the mixing time is of order $N^{2}$.

Résumé

Nous analysons temps de mélange pour le processus d’exclusion simple sur un cercle de $N$ sommets, avec un nombre de particules $k(N)$ qui tend vers l’infini avec $N$, et partant de la pire configuration initiale possible. Nous étudions également le cas d’une configuration initiale typique. Nous montrons que le temps de mélange est asymptotiquement équivalent $(8\pi^{2})^{-1}N^{2}\log k$, pour la pire condition initiale, et que la fenêtre de cutoff est d’ordre $N^{2}$. Dans le cas d’une condition initiale typique nous montrons qu’il n’y a pas de cutoff et que le temps de mélange est d’ordre $N^{2}$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1402-1437.

Dates
Received: 4 January 2016
Revised: 6 April 2016
Accepted: 14 April 2016
First available in Project Euclid: 21 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624046

Digital Object Identifier
doi:10.1214/16-AIHP759

Mathematical Reviews number (MathSciNet)
MR3689972

Zentralblatt MATH identifier
1379.82023

Subjects
Primary: 82D60: Polymers 60K37: Processes in random environments 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Markov chains Mixing time Particle systems Cutoff Window

Citation

Lacoin, Hubert. The simple exclusion process on the circle has a diffusive cutoff window. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1402--1437. doi:10.1214/16-AIHP759. https://projecteuclid.org/euclid.aihp/1500624046


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