The Annals of Applied Probability

Eigenvalue Bounds on Convergence to Stationarity for Nonreversible Markov Chains, with an Application to the Exclusion Process

James Allen Fill

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Abstract

We extend recently developed eigenvalue bounds on mixing rates for reversible Markov chains to nonreversible chains. We then apply our results to show that the $d$-particle simple exclusion process corresponding to clockwise walk on the discrete circle $\mathbf{Z}_p$ is rapidly mixing when $d$ grows with $p$. The dense case $d = p/2$ arises in a Poisson blockers problem in statistical mechanics.

Article information

Source
Ann. Appl. Probab. Volume 1, Number 1 (1991), 62-87.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177005981

Digital Object Identifier
doi:10.1214/aoap/1177005981

Mathematical Reviews number (MathSciNet)
MR1097464

Zentralblatt MATH identifier
0726.60069

JSTOR
links.jstor.org

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 15A42: Inequalities involving eigenvalues and eigenvectors

Keywords
Markov chains reversibility variation distance rates of convergence rapid mixing Poincare inequality Cheege's inequality chi-square distance interacting particle systems exclusion process Poisson blockers

Citation

Fill, James Allen. Eigenvalue Bounds on Convergence to Stationarity for Nonreversible Markov Chains, with an Application to the Exclusion Process. Ann. Appl. Probab. 1 (1991), no. 1, 62--87. doi:10.1214/aoap/1177005981. https://projecteuclid.org/euclid.aoap/1177005981.


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