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Dobrushin Conditions and Systematic Scan

Published online by Cambridge University Press:  01 November 2008

MARTIN DYER ,
LESLIE ANN GOLDBERG  and
MARK JERRUM
MARTIN DYER
Affiliation:
School of Computing, University of Leeds, Leeds LS2 9JT, UK
LESLIE ANN GOLDBERG
Affiliation:
Department of Computer Science, University of Liverpool, Liverpool L69 3BX, UK
MARK JERRUM
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London Mile End Road, London E1 4NS, UK
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Abstract

We consider Glauber dynamics on finite spin systems. The mixing time of Glauber dynamics can be bounded in terms of the influences of sites on each other. We consider three parameters bounding these influences: α, the total influence on a site, as studied by Dobrushin; α′, the total influence of a site, as studied by Dobrushin and Shlosman; and α″, the total influence of a site in any given context, which is related to the path-coupling method of Bubley and Dyer. It is known that if any of these parameters is less than 1 then random-update Glauber dynamics (in which a randomly chosen site is updated at each step) is rapidly mixing. It is also known that the Dobrushin condition α < 1 implies that systematic-scan Glauber dynamics (in which sites are updated in a deterministic order) is rapidly mixing. This paper studies two related issues, primarily in the context of systematic scan: (1) the relationship between the parameters α, α′ and α″, and (2) the relationship between proofs of rapid mixing using Dobrushin uniqueness (which typically use analysis techniques) and proofs of rapid mixing using path coupling. We use matrix balancing to show that the Dobrushin–Shlosman condition α′ < 1 implies rapid mixing of systematic scan. An interesting question is whether the rapid mixing results for scan can be extended to the α = 1 or α′ = 1 case. We give positive results for the rapid mixing of systematic scan for certain α = 1 cases. As an application, we show rapid mixing of systematic scan (for any scan order) for heat-bath Glauber dynamics for proper q-colourings of a degree-Δ graph G when q ≥ 2Δ.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 6 , November 2008 , pp. 761 - 779
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Aldous, D. (1982) Some inequalities for reversible Markov chains. J. London Math. Soc. (2) 25 564–576.CrossRefGoogle Scholar
[2]Bubley, R. and Dyer, M. E. (1997) Path coupling: A technique for proving rapid mixing in Markov chains. In Proc. 38th Annual Symposium on Foundations of Computer Science (FOCS '97), pp. 223–231.CrossRefGoogle Scholar
[3]Cho, G. E. and Meyer, C. D. (2000) Markov chain sensitivity measured by mean first passage times. Linear Algebra Appl. 316 21–28.CrossRefGoogle Scholar
[4]Cho, G. E. and Meyer, C. D. (2001) Comparison of perturbation bounds for the stationary distribution of a Markov chain. Linear Algebra Appl. 335 137–150.CrossRefGoogle Scholar
[5]Diaconis, P. and Saloff-Coste, L. (1993) Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696–730.CrossRefGoogle Scholar
[6]Diaconis, P. and Stroock, D. (1991) Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 36–61.CrossRefGoogle Scholar
[7]Dobrushin, R. L. (1970) Prescribing a system of random variables by conditional distributions. Theory Probab. Appl. 15 458–486.CrossRefGoogle Scholar
[8]Dobrushin, R. L. and Shlosman, S. B. (1985) Constructive criterion for the uniqueness of a Biggs field. In Statistical Mechanics and Dynamical Systems (Fritz, J., Jaffe, A. and Szasz, D., eds), Birkhäuser, Boston, pp. 347–370.CrossRefGoogle Scholar
[9]Dyer, M., Goldberg, L. A. and Jerrum, M. (2006) Systematic scan for sampling colourings. Ann. Appl. Probab., 16 185–230.CrossRefGoogle Scholar
[10]Dyer, M., Goldberg, L. A., Jerrum, M. and Martin, R. (2006) Markov chain comparison. Probability Surveys 3 89–111.CrossRefGoogle Scholar
[11]Dyer, M. and Greenhill, C. (1999) Random walks on combinatorial objects. In Surveys in Combinatorics (Lamb, J. D. and Preece, D. A., eds), Vol. 267 of London Mathematical Society Lecture Notes, Cambridge University Press, pp. 101–136.Google Scholar
[12]Föllmer, H. (1982) A covariance estimate for Gibbs measures. J. Funct. Anal. 46 387–395.CrossRefGoogle Scholar
[13]Lovász, L. and Winkler, P. (1995) Mixing of random walks and other diffusions on a graph. In Surveys in Combinatorics (Rowlinson, P., ed.), Vol. 218 of London Mathematical Society Lecture Notes, pp. 119–154.CrossRefGoogle Scholar
[14]Martin, R. and Randall, D. (2000) Sampling adsorbing staircase walks using a new Markov chain decomposition method. In Proc. 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2000), pp. 492–502.CrossRefGoogle Scholar
[15]Pedersen, K. Personal communication.Google Scholar
[16]Salas, J. and Sokal, A. D. (1997) Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. J. Statist. Phys. 86 551–579.CrossRefGoogle Scholar
[17]Seneta, E. (2006) Non-Negative Matrices and Markov Chains, 2nd edn, Springer, New York.Google Scholar
[18]Simon, B. (1993) The Statistical Mechanics of Lattice Gases, Princeton University Press.Google Scholar
[19]Sinclair, A. (1992) Improved bounds for mixing rates of Markov chains and multicommodity flow. Combin. Probab. Comput. 1 351–370.CrossRefGoogle Scholar
[20]Sokal, A. (2001) A personal list of unsolved problems concerning lattice gases and antiferromagnetic Potts models. Markov Processes and Related Fields, 7 21–38.Google Scholar
[21]Weitz, D. (2005) Combinatorial criteria for uniqueness of Gibbs measures. Random Struct. Alg. 27 445–475.CrossRefGoogle Scholar