Dobrushin conditions and systematic scan

  • Authors:

  • Martin Dyer;Leslie Ann Goldberg;Mark Jerrum

  • Affiliations:

  • School of Computing, University of Leeds, Leeds, UK;Department of Computer Science, University of Warwick, Coventry, UK;School of Informatics, University of Edinburgh, Edinburgh, UK

  • Venue:
  • APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
  • Year:
  • 2006

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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 αα, 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′ 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.