Electronic Journal of Probability

Mixing times for the simple exclusion process in ballistic random environment

Dominik Schmid

Full-text: Open access

Abstract

We consider the exclusion process on segments of the integers in a site-dependent random environment. We assume to be in the ballistic regime in which a single particle has positive linear speed. Our goal is to study the mixing time of the exclusion process when the number of particles is linear in the size of the segment. We investigate the order of the mixing time depending on the support of the environment distribution. In particular, we prove for nestling environments that the order of the mixing time is different than in the case of a single particle.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 22, 25 pp.

Dates
Received: 18 June 2018
Accepted: 2 March 2019
First available in Project Euclid: 21 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1553155302

Digital Object Identifier
doi:10.1214/19-EJP286

Mathematical Reviews number (MathSciNet)
MR3933201

Zentralblatt MATH identifier
1412.60137

Subjects
Primary: 60K37: Processes in random environments 60J27: Continuous-time Markov processes on discrete state spaces

Keywords
exclusion process mixing time random environment

Rights
Creative Commons Attribution 4.0 International License.

Citation

Schmid, Dominik. Mixing times for the simple exclusion process in ballistic random environment. Electron. J. Probab. 24 (2019), paper no. 22, 25 pp. doi:10.1214/19-EJP286. https://projecteuclid.org/euclid.ejp/1553155302


Export citation

References

  • [1] Itai Benjamini, Noam Berger, Christopher Hoffman, and Elchanan Mossel, Mixing times of the biased card shuffling and the asymmetric exclusion process, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3013–3029.
  • [2] R. A. Blythe, M. R. Evans, F. Colaiori, and F. H. L. Essler, Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra, J. Phys. A 33 (2000), no. 12, 2313–2332.
  • [3] Nina Gantert and Thomas Kochler, Cutoff and mixing time for transient random walks in random environments, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 1, 449–484.
  • [4] Jonathan Hermon and Richard Pymar, The exclusion process mixes (almost) faster than independent particles, preprint. arXiv:1808.10846
  • [5] Paul Jung, Extremal reversible measures for the exclusion process, J. Statist. Phys. 112 (2003), no. 1-2, 165–191.
  • [6] H. Kesten, M. V. Kozlov, and F. Spitzer, A limit law for random walk in a random environment, Compositio Math. 30 (1975), 145–168.
  • [7] Cyril Labbé and Hubert Lacoin, Cutoff phenomenon for the asymmetric simple exclusion process and the biased card shuffling, Ann. Probab. (to appear). arXiv:1610.07383
  • [8] Cyril Labbé and Hubert Lacoin, Mixing time and cutoff for the weakly asymmetric simple exclusion process, preprint. arXiv:1805.12213
  • [9] Hubert Lacoin, Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion, Ann. Probab. 44 (2016), no. 2, 1426–1487.
  • [10] C. Landim, A. Milanés, and S. Olla, Stationary and nonequilibrium fluctuations in boundary driven exclusion processes, Markov Process. Related Fields 14 (2008), no. 2, 165–184.
  • [11] David A. Levin and Yuval Peres, Mixing of the exclusion process with small bias, J. Stat. Phys. 165 (2016), no. 6, 1036–1050.
  • [12] David A. Levin and Yuval Peres, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2017, Second edition of [MR2466937], With contributions by Elizabeth L. Wilmer, With a chapter on “Coupling from the past” by James G. Propp and David B. Wilson.
  • [13] Thomas M. Liggett, Interacting particle systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276, Springer-Verlag, New York, 1985.
  • [14] Thomas M. Liggett, Stochastic interacting systems: contact, voter and exclusion processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 324, Springer-Verlag, Berlin, 1999.
  • [15] Sarah Miracle and Amanda Pascoe Streib, Rapid mixing of $k$-class biased permutations, LATIN 2018: Theoretical informatics, Lecture Notes in Comput. Sci., vol. 10807, Springer, Cham, 2018, pp. 820–834.
  • [16] Roberto Imbuzeiro Oliveira, Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk, Ann. Probab. 41 (2013), no. 2, 871–913.
  • [17] Yuval Peres and Peter Winkler, Can extra updates delay mixing?, Comm. Math. Phys. 323 (2013), no. 3, 1007–1016.
  • [18] Fred Solomon, Random walks in a random environment, Ann. Probability 3 (1975), 1–31.
  • [19] David Bruce Wilson, Mixing times of Lozenge tiling and card shuffling Markov chains, Ann. Appl. Probab. 14 (2004), no. 1, 274–325.
  • [20] Ofer Zeitouni, Random walks in random environment, Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1837, Springer, Berlin, 2004, pp. 189–312.