Cluster Expansions of Small Contours in Abstract Pirogov - Sinai Models

M. Zahradnik

2002, v.8, Issue 3, 383-441


We develop the Pirogov - Sinai theory for "abstract Pirogov - Sinai models" where configuration is already represented as a system of matching compatible contours. The Hamiltonian is then expressed in terms of contour energies and an additional external field, the latter acting not on single spins but more generally on their "clusters", in the space ("colored" by various "local ground states") outside of contours. This general model is formulated in such a way that i) its general structure does not change when cluster expansion of its "smallest possible" contours resp. contour systems is applied; ii) it includes, after suitable preparation, many interesting lattice spin models, notably the reformulation of the Ising Kac ferromagnet studied in the paper [A. Bovier and M. Zahradnik, Cluster Expansions and Pirogov - Sinai Theory for Long Range Spin Systems, Markov Processes Relat. Fields, 2002, v. 8, N 3, 443-478]. In fact the result of the latter paper - a model with expanded restricted ensembles living outside of suitably defined contours - is the main testing example for the usefulness of this general method, further elaborating and simplifying the method of the paper [M. Zahradnik, A short course in the Pirogov - Sinai theory, Rendiconti Mat. Appl., 1997, v. 18, N 7, 411-486].

Keywords: low temperature Gibbs states,contours,cluster expansion,contourfunctional,abstract Pirogov - Sinai model with cluster field,Pirogov - Sinai theory


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