Specifications and Martin boundaries for \(\mathcal P(\phi)_2\)-random fields (Q1819829)

It is shown that \(\mathcal P(\phi)_2\)-Gibbs states in the sense of \textit{F. Guerra}, \textit{J. Rosen} and \textit{B. Simon} [``Euclidean quantum field theory as classical statistical mechanics.'' Ann. Math. 101, 111--189, 191--259 (1975) and ``Boundary conditions in the \(\mathcal P(\phi)_2\) Euclidean field theory.'' Ann. Inst. Henri Poincaré, Sect. A 15, 231--334 (1976)] are given by a specification. The construction of the specification is based on finding a proper version of the interaction density given by the polynomial \(\mathcal P\). The existence of this version follows from the fact that all powers of the solution of a Dirichlet problem for an open bounded set \(U\) with boundary data given by a distribution are integrable on \(U\). As a consequence the Martin boundary theory for specifications can be applied to \(\mathcal P(\phi)_2\)-random fields. It follows that any \(\mathcal P(\phi)_2\)-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 cutoff \(\mathcal P(\phi)_2\)-measure has a representation as an integral over harmonic functions.