EXISTENCE AND UNIQUENESS OF DLR MEASURES FOR UNBOUNDED SPIN SYSTEMS

A system of random variables(spins) Sx,x∈ℤ v, taking on values in ℝ is considered. Conditional probabilities for the joint distributions of a finite number of spins are prescribed; a DLR measure is then a process on the random variables which is consistent with the assigned conditional probabilities [1,2]. A case of physical interest both in Statistical Mechanics and in the lattice approximation to Quantum Field Theory is considered for which the spins interact pairwise via a potential JxySxSy, Jxy∈ℝ and via a self-interaction F(Sx), which, as |Sx|→∞, diverges at least quadratically [3]. By use of a technique introduced in [2] it is proven that the set {Mathematical expression} is a compact (in the local weak topology, Def. 1.1) non-void Choquet simplex [4]. Sufficient conditions are then given in order to obtain the measures in {Mathematical expression} as limits of Gibbs measures for finitely many spins in a wide class of boundary conditions, Theorem 1.2. Uniqueness in {Mathematical expression} is then discussed by means of a theorem by Dobrušin [2] and a sufficient condition for unicity is obtained which can be physically interpreted as a mean field condition [5]. Therefore the mean field temperature is rigorously proven to be an upper bound for the critical temperature. © 1978 Springer-Verlag.