The Annals of Applied Probability

Matrix norms and rapid mixing for spin systems

Martin Dyer, Leslie Ann Goldberg, and Mark Jerrum

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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.

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Ann. Appl. Probab., Volume 19, Number 1 (2009), 71-107.

First available in Project Euclid: 20 February 2009

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Primary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 68W20: Randomized algorithms 68W40: Analysis of algorithms [See also 68Q25] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Matrix norms rapid mixing Markov chains spin systems


Dyer, Martin; Goldberg, Leslie Ann; Jerrum, Mark. Matrix norms and rapid mixing for spin systems. Ann. Appl. Probab. 19 (2009), no. 1, 71--107. doi:10.1214/08-AAP532.

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