Specifications and Martin boundaries for \(\mathcal P(\phi)_2\)-random fields
type: Item
instance of: scholarly article
title: Specifications and Martin boundaries for \(\mathcal P(\phi)_2\)-random fields
zbMATH Open document ID: 0614.60068
DOI: 10.1007/BF01210928
published in: Communications in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 60J50
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 82B05
zbMATH DE Number: 3994686
zbMATH Keywords: interaction density
zbMATH Keywords: Martin boundary theory
zbMATH Keywords: extreme Gibbs states
zbMATH Keywords: boundary condition
author: Michael Röckner
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01210928
OpenAlex ID: W2248639248
description: scientific article; zbMATH DE number 3994686