## The Annals of Probability

### Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk

Roberto Imbuzeiro Oliveira

#### Abstract

We prove an upper bound for the $\varepsilon$-mixing time of the symmetric exclusion process on any graph $G$, with any feasible number of particles. Our estimate is proportional to ${\mathsf{T}}_{\mathsf{RW}(G)}\ln(|V|/\varepsilon )$, where $|V|$ is the number of vertices in $G$, and ${\mathsf{T}}_{\mathsf{RW}(G)}$ is the $1/4$-mixing time of the corresponding single-particle random walk. This bound implies new results for symmetric exclusion on expanders, percolation clusters, the giant component of the Erdös–Rényi random graph and Poisson point processes in $\mathbb{R}^{d}$. Our technical tools include a variant of Morris’s chameleon process.

#### Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 871-913.

Dates
First available in Project Euclid: 8 March 2013

https://projecteuclid.org/euclid.aop/1362750945

Digital Object Identifier
doi:10.1214/11-AOP714

Mathematical Reviews number (MathSciNet)
MR3077529

Zentralblatt MATH identifier
1274.60242

#### Citation

Oliveira, Roberto Imbuzeiro. Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk. Ann. Probab. 41 (2013), no. 2, 871--913. doi:10.1214/11-AOP714. https://projecteuclid.org/euclid.aop/1362750945

#### References

• [1] Aldous, D. and Fill, J. A. Reversible Markov Chains and random walks on graphs. Book draft. Available at http://www.stat.berkeley.edu/~aldous/RWG/book.html.
• [2] Alon, N. and Spencer, J. H. (2000). The Probabilistic Method, 2nd ed. Wiley, New York.
• [3] Andjel, E. D. (1988). A correlation inequality for the symmetric exclusion process. Ann. Probab. 16 717–721.
• [4] Benjamini, I. and Mossel, E. (2003). On the mixing time of a simple random walk on the super critical percolation cluster. Probab. Theory Related Fields 125 408–420.
• [5] Caputo, P. and Faggionato, A. (2007). Isoperimetric inequalities and mixing time for a random walk on a random point process. Ann. Appl. Probab. 17 1707–1744.
• [6] Caputo, P., Liggett, T. M. and Richthammer, T. (2010). Proof of Aldous’ spectral gap conjecture. J. Amer. Math. Soc. 23 831–851.
• [7] Cooper, C., Frieze, A. and Radzik, T. (2009). Multiple random walks and interacting particle systems. In Automata, Languages and Programming. Part II. Lecture Notes in Computer Science 5556 399–410. Springer, Berlin.
• [8] Diaconis, P. and Saloff-Coste, L. (1993). Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696–730.
• [9] Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750.
• [10] Dieker, A. B. (2010). Interlacings for random walks on weighted graphs and the interchange process. SIAM J. Discrete Math. 24 191–206.
• [11] Fountoulakis, N. and Reed, B. A. (2008). The evolution of the mixing rate of a simple random walk on the giant component of a random graph. Random Structures Algorithms 33 68–86.
• [12] Lee, T.-Y. and Yau, H.-T. (1998). Logarithmic Sobolev inequality for some models of random walks. Ann. Probab. 26 1855–1873.
• [13] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
• [14] Liggett, T. M. (1974). A characterization of the invariant measures for an infinite particle system with interactions. II. Trans. Amer. Math. Soc. 198 201–213.
• [15] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
• [16] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 324. Springer, Berlin.
• [17] Montenegro, R. and Tetali, P. (2006). Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1 x+121.
• [18] Morris, B. (2006). Spectral gap for the zero range process with constant rate. Ann. Probab. 34 1645–1664.
• [19] Morris, B. (2006). The mixing time for simple exclusion. Ann. Appl. Probab. 16 615–635.
• [20] Morris, B. (2009). Improved mixing time bounds for the Thorp shuffle and $L$-reversal chain. Ann. Probab. 37 453–477.
• [21] Morris, B. and Peres, Y. (2005). Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 245–266.
• [22] Pete, G. (2008). A note on percolation on $\mathbb{Z}^{d}$: Isoperimetric profile via exponential cluster repulsion. Electron. Commun. Probab. 13 377–392.
• [23] Yau, H.-T. (1997). Logarithmic Sobolev inequality for generalized simple exclusion processes. Probab. Theory Related Fields 109 507–538.