Specifications and Martin boundaries forP(Φ)2-random fields

Abstract

It is shown thatP(Φ)2-Gibbs states in the sense of Guerra, Rosen and Simon are given by a specification. The construction of the specification is based on finding a proper version of the interaction density given by the polynomialP. The existence of this version follows from the fact that all powers of the solution of a Dirichlet problem for an open bounded setU with boundary data given by a distribution are integrable onU. As a consequence the Martin boundary theory for specifications can be applied toP(Φ)2-random fields. It follows that anyP(Φ)2-Gibbs state can be represented in terms of extreme Gibbs states. In certain cases the extreme Gibbs states are characterized in terms of harmonic functions. It follows, in particular, that for any given boundary condition introduced so far the associated cutoffP(Φ)2-measure has a representation as an integral over harmonic functions.