The Annals of Probability

The cutoff profile for the simple exclusion process on the circle

Hubert Lacoin

Full-text: Open access

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

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

Article information

Source
Ann. Probab., Volume 44, Number 5 (2016), 3399-3430.

Dates
Received: March 2015
Revised: July 2015
First available in Project Euclid: 21 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1474462102

Digital Object Identifier
doi:10.1214/15-AOP1053

Mathematical Reviews number (MathSciNet)
MR3551201

Zentralblatt MATH identifier
06653521

Subjects
Primary: 37L60: Lattice dynamics [See also 37K60] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Markov chains mixing time particle systems cutoff profile

Citation

Lacoin, Hubert. The cutoff profile for the simple exclusion process on the circle. Ann. Probab. 44 (2016), no. 5, 3399--3430. doi:10.1214/15-AOP1053. https://projecteuclid.org/euclid.aop/1474462102


Export citation

References

  • [1] Azuma, K. (1967). Weighted sums of certain dependent random variables. Tôhoku Math. J. (2) 19 357–367.
  • [2] Brown, B. M. (1971). Martingale central limit theorems. Ann. Math. Statist. 42 59–66.
  • [3] Caputo, P., Liggett, T. M. and Richthammer, T. (2010). Proof of Aldous’ spectral gap conjecture. J. Amer. Math. Soc. 23 831–851.
  • [4] Diaconis, P., Fill, J. A. and Pitman, J. (1992). Analysis of top to random shuffles. Combin. Probab. Comput. 1 135–155.
  • [5] Diaconis, P., Graham, R. L. and Morrison, J. A. (1990). Asymptotic analysis of a random walk on a hypercube with many dimensions. Random Structures Algorithms 1 51–72.
  • [6] Diaconis, P. and Shahshahani, M. (1987). Time to reach stationarity in the Bernoulli–Laplace diffusion model. SIAM J. Math. Anal. 18 208–218.
  • [7] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Springer, Berlin.
  • [8] Lacoin, H. (2015). Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion. Ann. Probab. 44 1426–1487.
  • [9] Lacoin, H. (2016). The simple exclusion process on the circle has a diffusive cutoff window. Preprint. Available at arXiv:1401.7296.
  • [10] Lacoin, H. and Leblond, R. (2011). Cutoff phenomenon for the simple exclusion process on the complete graph. ALEA Lat. Am. J. Probab. Math. Stat. 8 285–301.
  • [11] Lee, T.-Y. and Yau, H.-T. (1998). Logarithmic Sobolev inequality for some models of random walks. Ann. Probab. 26 1855–1873.
  • [12] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [13] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 324. Springer, Berlin.
  • [14] Liggett, T. M. (2005). Interacting Particle Systems. Springer, Berlin.
  • [15] Matthews, P. (1988). A strong uniform time for random transpositions. J. Theoret. Probab. 1 411–423.
  • [16] Morris, B. (2006). The mixing time for simple exclusion. Ann. Appl. Probab. 16 615–635.
  • [17] Oliveira, R. I. (2013). Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk. Ann. Probab. 41 871–913.
  • [18] Rost, H. (1981). Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk. Z. Wahrsch. Verw. Gebiete 58 41–53.
  • [19] Wilson, D. B. (2004). Mixing times of Lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14 274–325.