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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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.

Article information

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

Dates
First available in Project Euclid: 20 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1235140333

Digital Object Identifier
doi:10.1214/08-AAP532

Mathematical Reviews number (MathSciNet)
MR2498672

Zentralblatt MATH identifier
1166.15015

Subjects
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

Keywords
Matrix norms rapid mixing Markov chains spin systems

Citation

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. https://projecteuclid.org/euclid.aoap/1235140333


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