Dobrushin Conditions and Systematic Scan

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.