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Matrix norms and rapid mixing for spin systems

By M. Dyer, L.A. Goldberg and M. Jerrum

Abstract

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

Publisher: Institute of Mathematical Statistics
Year: 2009
OAI identifier: oai:eprints.whiterose.ac.uk:7982

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