ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 $$\mathfrak{E} = \{ v is DLR|\exists c(v), \mathop {sup}\limits_{x \in \mathbb{Z}^v } \int {v(dS_x )} |S_x | 〈 c(v)\} $$ 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00533602
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