Advertisement

Dobrushin Conditions for Systematic Scan with Block Dynamics

  • Kasper Pedersen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)

Abstract

We study the mixing time of systematic scan Markov chains on finite spin systems. It is known that, in a single site setting, the mixing time of systematic scan can be bounded in terms of the influences sites have on each other. We generalise this technique for bounding the mixing time of systematic scan to block dynamics, a setting in which a set of sites are updated simultaneously. In particular we present a parameter α, representing the maximum influence on any site, and show that if α< 1 then the corresponding systematic scan Markov chain mixes rapidly. We use this method to prove O(logn) mixing of a systematic scan for proper q-colourings of a general graph with maximum vertex-degree Δ whenever q ≥ 2Δ. We also apply the method to improve the number of colours required in order to obtain mixing in O(logn) scans for a systematic scan colouring of trees.

Keywords

Markov Chain Transition Matrix Spin System Gibbs Measure General Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dyer, M., Goldberg, L.A., Jerrum, M.: Dobrushin conditions and systematic scan. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds.) APPROX 2006 and RANDOM 2006. LNCS, vol. 4110, pp. 327–338. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Dyer, M., Goldberg, L.A., Jerrum, M.: Systematic scan and sampling colourings. Annals of Applied Probability 16(1), 185–230 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bordewich, M., Dyer, M., Karpinski, M.: Stopping times, metrics and approximate counting. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 108–119. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Dobrushin, R.L., Shlosman, S.B.: Constructive criterion for the uniqueness of Gibbs field. In: Statistical mechanics and dynamical systems, Progress in Physics, vol. 10, pp. 371–403. Birkhäuser, Boston (1985)Google Scholar
  5. 5.
    Weitz, D.: Mixing in Time and Space for Discrete Spin Systems. PhD thesis, University of California, Berkley (2004)Google Scholar
  6. 6.
    Weitz, D.: Combinatorial criteria for uniqueness of Gibbs measures. Random Structures and Algorithms 27(4), 445–475 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dyer, M., Sinclair, A., Vigoda, E., Weitz, D.: Mixing in time and space for lattice spin systems: A combinatorial view. Random Structures and Algorithms 24(4), 461–479 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Theory Prob. and its Appl. 15, 458–486 (1970)zbMATHCrossRefGoogle Scholar
  9. 9.
    Aldous, D.J: Random walks on finite groups and rapidly mixing markov chains. In: Séminaire de probabilités XVII, pp. 243–297. Springer, Heidelberg (1983)Google Scholar
  10. 10.
    Simon, B.: The Statistical Mechanics of Lattice Gases. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  11. 11.
    Pedersen, K.: Dobrushin conditions for systematic scan with block dynamics. arXiv:math.PR/0703461 (2007)Google Scholar
  12. 12.
    Vigoda, E.: Improved bounds for sampling colourings. J. Math. Phys  (2000)Google Scholar
  13. 13.
    Hayes, T.P.: A simple condition implying rapid mixing of single-site dynamics on spin systems. In: Proc. 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 39–46. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  14. 14.
    Dyer, M., Goldberg, L.A., Jerrum, M.: Matrix norms and rapid mixing for spin systems. ArXiv math.PR/0702744  (2006)Google Scholar
  15. 15.
    Martinelli, F., Sinclair, A., Weitz, D.: Glauber dynamics on trees: Boundary conditions and mixing time. Communications in Mathematical Physics 250(2), 301–334 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kenyon, C., Mossel, E., Peres, Y.: Glauber dynamics on trees and hyperbolic graphs. In: Proc. 42nd Annual IEEE Symposium on Foundations of Computer Science, pp. 568–578. IEEE Computer Society Press, Los Alamitos (2001)Google Scholar
  17. 17.
    Föllmer, H.: A covariance estimate for Gibbs measures. J. Funct. Analys. 46, 387–395 (1982)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Kasper Pedersen
    • 1
  1. 1.Department of Computer Science, University of LiverpoolUK

Personalised recommendations