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.


Markov Chain Transition Matrix Spin System Full Version Glauber Dynamic 
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  1. 1.
    Aldous, D.: Some inequalities for reversible Markov chains. J. London Math. Society 25(2), 564–576 (1982)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bubley, R., Dyer, M.: Path coupling: a technique for proving rapid mixing in Markov chains. In: FOCS, vol. 38, pp. 223–231 (1997)Google Scholar
  3. 3.
    Cho, G.E., Meyer, C.D.: Markov chain sensitivity measured by mean first passage times. Linear Algebra and its Applications 316(1–3), 21–28 (2000)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cho, G.E., Meyer, C.D.: Comparison of perturbation bounds for the stationary distribution of a Markov chain. Linear Algebra and its Applications 335(1–3), 137–150 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Diaconis, P., Saloff-Coste, L.: Comparison theorems for reversible Markov chains. Annals of Applied Probability 3, 696–730 (1993)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Diaconis, P., Stroock, D.: Geometric bounds for eigenvalues of Markov chains. Annals of Applied Probability 1, 36–61 (1991)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Theory Prob. and its Appl. 15, 458–486 (1970)MATHCrossRefGoogle Scholar
  8. 8.
    Dobrushin, R.L., Shlosman, S.B.: Constructive criterion for the uniqueness of a Biggs field. In: Fritz, J., Jaffe, A., Szasz, D. (eds.) Statistical mechanics and dynamical systems, pp. 347–370. Birkhauser, Boston (1985)Google Scholar
  9. 9.
    Dyer, M., Goldberg, L.A., Jerrum, M.: Systematic scan for sampling colourings. Annals of Applied Probability (to appear)Google Scholar
  10. 10.
    Dyer, M., Goldberg, L.A., Jerrum, M., Martin, R.: Markov chain comparison (preprint, 2004)Google Scholar
  11. 11.
    Dyer, M., Greenhill, C.: Random walks on combinatorial objects. In: Lamb, J.D., Preece, D.A. (eds.) Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 267, pp. 101–136. Cambridge University Press, Cambridge (1999)Google Scholar
  12. 12.
    Föllmer, H.: A covariance estimate for Gibbs measures. J. Funct. Analys. 46, 387–395 (1982)MATHCrossRefGoogle Scholar
  13. 13.
    Lovśz, L., Winkler, P.: Mixing of random walks and other diffusions on a graph. In: Rowlinson, P. (ed.) Surveys in Combinatorics. London Math. Soc. Lecture Note Series, vol. 218, pp. 119–154 (1995)Google Scholar
  14. 14.
    Martin, R., Randall, D.: Sampling adsorbing staircase walks using a new Markov chain decomposition method. In: Proc. of the 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2000), pp. 492–502 (2000)Google Scholar
  15. 15.
    Pedersen, K.: Personal communicationGoogle Scholar
  16. 16.
    Salas, J., Sokal, A.D.: Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. J. Statistical Physics 86, 551–579 (1997)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Sinclair, A.: Improved bounds for mixing rates of Markov chains and multicommodity flow. Combinatorics, Probability and Computing 1, 351–370 (1992)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Simon, B.: The Statistical Mechanics of Lattice Gases. Princeton University Press, Princeton (1993)MATHGoogle Scholar
  19. 19.
    Sokal, A.: A personal list of unsolved problems concerning lattice gases and antiferromagnetic potts models. Markov Processes and Related Fields (to appear)Google Scholar
  20. 20.
    Weitz, D.: Combinatorial Criteria for Uniqueness of Gibbs Measures. Random Structures and Algorithms (to appear, 2005)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Martin Dyer
    • 1
  • Leslie Ann Goldberg
    • 2
  • Mark Jerrum
    • 3
  1. 1.School of ComputingUniversity of LeedsLeedsUK
  2. 2.Department of Computer ScienceUniversity of WarwickCoventryUK
  3. 3.School of InformaticsUniversity of EdinburghEdinburghUK

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