The Annals of Applied Probability

Geometric Bounds for Eigenvalues of Markov Chains

Persi Diaconis and Daniel Stroock

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Abstract

We develop bounds for the second largest eigenvalue and spectral gap of a reversible Markov chain. The bounds depend on geometric quantities such as the maximum degree, diameter and covering number of associated graphs. The bounds compare well with exact answers for a variety of simple chains and seem better than bounds derived through Cheeger-like inequalities. They offer improved rates of convergence for the random walk associated to approximate computation of the permanent.

Article information

Source
Ann. Appl. Probab., Volume 1, Number 1 (1991), 36-61.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177005980

Digital Object Identifier
doi:10.1214/aoap/1177005980

Mathematical Reviews number (MathSciNet)
MR1097463

Zentralblatt MATH identifier
0731.60061

JSTOR
links.jstor.org

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60C05: Combinatorial probability

Keywords
Markov chains eigenvalues random walk

Citation

Diaconis, Persi; Stroock, Daniel. Geometric Bounds for Eigenvalues of Markov Chains. Ann. Appl. Probab. 1 (1991), no. 1, 36--61. doi:10.1214/aoap/1177005980. https://projecteuclid.org/euclid.aoap/1177005980


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