A remark on Dobrushin's uniqueness theorem

Abstract

Ten years ago, Dobrushin [1] proved a beautiful result showing that under suitable hypotheses, a statistical mechanical lattice system interaction has a unique equilibrium state. In particular, there is no long range order, etc. see [6,7] for related material, Israel [4] for analyticity results and Gross [3] for falloff of correlations. There does not appear to have been systematic attempts to obtain very good estimates on precisely when Dobrushin's hypotheses hold, except for certain spin models [6,4]. Our purpose here is to note that with one simple device one can obtain extremely good estimates which are fairly close to optimal. Let Ω be a fixed compact space (single spin configuration space), dμ0 a probability measure on Ω and for each α e Z, Ωα a copy of Ω. For X a finite subset of TD', let Ω — X Ωα. An interaction is an assignment of a continuous function, ΦpO, on Ω to each finite XcTL. While it is not necessary for Dobrushin's theorem, it is convient notationally to suppose Φ translation covariant in the obvious sense.