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Matrix norms and rapid mixing for spin systems

Dyer, M., Goldberg, L.A. and Jerrum, M. (2009) Matrix norms and rapid mixing for spin systems. Annals of Applied Probability, 19 (1). pp. 71-107. ISSN 1050-5164

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Abstract

We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is 0 (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for coloring various classes of graphs. We give a general method for estimating a norm of a symmetric nonregular matrix. This leads to improved mixing times for any class of graphs which is hereditary and sufficiently sparse including several classes of degree-bounded graphs such as nonregular graphs, trees, planar graphs and graphs with given tree-width and genus.

Item Type: Article
Copyright, Publisher and Additional Information: © Institute of Mathematical Statistics, 2009. Reproduced in accordance with the publisher's self-archiving pollicy.
Keywords: Matrix norms, rapid mixing, Markov chains, spin systems
Institution: The University of Leeds
Academic Units: The University of Leeds > Faculty of Engineering (Leeds) > School of Computing (Leeds)
Depositing User: Miss Jamie Grant
Date Deposited: 20 Mar 2009 14:13
Last Modified: 06 Jun 2014 23:16
Published Version: http://dx.doi.org/10.1214/08-AAP532
Status: Published
Publisher: Institute of Mathematical Statistics
Refereed: Yes
Identification Number: 10.1214/08-AAP532
URI: http://eprints.whiterose.ac.uk/id/eprint/7982

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