Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Hubert Lacoin

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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 $N^{2}$.


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Markov chains Mixing time Particle systems Cutoff Window


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.

Export citation


  • [1] D. Aldous. Random walks on finite groups and rapidly mixing Markov chains. In Seminar on Probability, XVII 243–297, 1983.
  • [2] D. Aldous and P. Diaconis. Shuffling cards and stopping times. Amer. Math. Monthly 93 (1986) 333–348.
  • [3] R. Basu, J. Hermon and Y. Peres. Characterization of cutoff for reversible Markov chains. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms 1774–1791. SIAM, Philadelphia, PA, 2015.
  • [4] P. Caputo, T. M. Liggett and T. Richthammer. Proof of Aldous’ spectral gap conjecture. J. Amer. Math. Soc. 23 (2010) 831–851.
  • [5] G. Y. Chen and L. Saloff-Coste. The cutoff phenomenon for ergodic Markov processes. Electron. J. Probab. 13 (2008) 26–78.
  • [6] P. Diaconis and L. Saloff-Coste. Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 (1993) 696–730.
  • [7] P. Diaconis and L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996) 695–750.
  • [8] P. Diaconis and M. Shahshahani. Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 (1981) 159–179.
  • [9] P. Diaconis and M. Shahshahani. Time to reach stationarity in the Bernoulli-Laplace diffusion model. SIAM J. Math. Anal. 18 (1987) 208–218.
  • [10] J. Ding, E. Lubetzky and Y. Peres. Total variation cutoff in birth-and-death chains. Probab. Theory Related Fields 146 (2010) 61–85.
  • [11] T. Fort. Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford, 1948.
  • [12] C. M. Fortuin, J. Ginibre and P. W. Kasteleyn. Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 (1971) 89–103.
  • [13] A. E. Holroyd. Some circumstances where extra updates can delay mixing. J. Stat. Phys. 145 (2011) 1649–1652.
  • [14] C. Kipnis and C. Landim. Scaling Limits of Interacting Particle Systems. Grund. fur Math. Wissen. 320. Springer, Berlin, 1999.
  • [15] H. Lacoin. Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion. Ann. Probab. 44 (2016) 1426–1487.
  • [16] H. Lacoin. The cutoff profile for the simple exclusion process on the circle. Ann. Probab. 44 (2016) 3399–3430.
  • [17] H. Lacoin and R. Leblond. The cutoff phenomenon for the simple exclusion process on the complete graph. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011) 285–301.
  • [18] H. Lacoin, F. Simenhaus and F. L. Toninelli. Zero-temperature stochastic Ising model in two dimension and anisotropic curve-shortening flow. J. Eur. Math. Soc. (JEMS) 16 (2014) 2557–2615.
  • [19] T. Y. Lee and H. T. Yau. Logarithmic Sobolev inequality for some models of random walks. Ann. Probab. 26 (1998) 1855–1873.
  • [20] D. Levin, Y. Peres and E. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, Providence, RI, 2009.
  • [21] T. M. Liggett. A characterization of the invariant measures for an infinite particle system with interaction. Trans. Amer. Math. Soc. 198 (1974) 201–213.
  • [22] T. M. Liggett. The stochastic evolution of infinite systems of interacting particles. In Ecole d’Eté de Probabilités de Saint-Flour VI (1976) 187–248. Lecture Notes in Mathematics 598, 1977.
  • [23] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grund. fur Math. Wissen. 324. Springer, Berlin, 1999.
  • [24] E. Lubetzky and A. Sly. Cutoff phenomena for random walks on random regular graphs. Duke Math. J. 153 (2010) 475–510.
  • [25] E. Lubetzky and A. Sly. Cutoff for the Ising model on the lattice. Invent. Math. 191 (2013) 719–755.
  • [26] B. Morris. The mixing time for simple exclusion. Ann. Appl. Probab. 16 (2006) 615–635.
  • [27] R. I. Oliveira. Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk. Ann. Probab. 41 (2013) 871–913.
  • [28] Y. Peres and P. Winkler. Can extra updates delay mixing. Comm. Math. Phys. 323 (2013) 1007–1016.
  • [29] J. Quastel. Diffusion of color in the simple exclusion process. Comm. Pure Appl. Math. 45 (1992) 623–679.
  • [30] H. Rost. Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk. Z. Wahrsch. Verw. Gebiete 58 (1981) 41–53.
  • [31] F. Sitzer. Interaction of Markov processes. Adv. Math. 5 (1970) 246–290.
  • [32] D. B. Wilson. Mixing times of Lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14 (2004) 274–325.
  • [33] H. T. Yau. Logarithmic Sobolev inequality for generalized simple exclusion processes. Probab. Theory Related Fields 109 (1997) 507–538.