Summary
A system of random variables(spins) S x,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 J xySxSy, Jxy∈ℝ and via a self-interaction F(S x), which, as ¦S x¦→∞, diverges at least quadratically [3].
By use of a technique introduced in [2] it is proven that the set
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 \(\mathfrak{E}\) as limits of Gibbs measures for finitely many spins in a wide class of boundary conditions, Theorem 1.2. Uniqueness in \(\mathfrak{E}\) 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.
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Cassandro, M., Olivieri, E., Pellegrinotti, A. et al. Existence and uniqueness of DLR measures for unbounded spin systems. Z. Wahrscheinlichkeitstheorie verw Gebiete 41, 313–334 (1978). https://doi.org/10.1007/BF00533602
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DOI: https://doi.org/10.1007/BF00533602