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June 2007 A general lower bound for mixing of single-site dynamics on graphs
Thomas P. Hayes, Alistair Sinclair
Ann. Appl. Probab. 17(3): 931-952 (June 2007). DOI: 10.1214/105051607000000104

Abstract

We prove that any Markov chain that performs local, reversible updates on randomly chosen vertices of a bounded-degree graph necessarily has mixing time at least Ω(n logn), where n is the number of vertices. Our bound applies to the so-called “Glauber dynamics” that has been used extensively in algorithms for the Ising model, independent sets, graph colorings and other structures in computer science and statistical physics, and demonstrates that many of these algorithms are optimal up to constant factors within their class. Previously, no superlinear lower bound was known for this class of algorithms. Though widely conjectured, such a bound had been proved previously only in very restricted circumstances, such as for the empty graph and the path. We also show that the assumption of bounded degree is necessary by giving a family of dynamics on graphs of unbounded degree with mixing time O(n).

Citation

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Thomas P. Hayes. Alistair Sinclair. "A general lower bound for mixing of single-site dynamics on graphs." Ann. Appl. Probab. 17 (3) 931 - 952, June 2007. https://doi.org/10.1214/105051607000000104

Information

Published: June 2007
First available in Project Euclid: 22 May 2007

zbMATH: 1125.60075
MathSciNet: MR2326236
Digital Object Identifier: 10.1214/105051607000000104

Subjects:
Primary: 60J10
Secondary: 60K35, 68W20, 68W25, 82C20

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.17 • No. 3 • June 2007
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