The Local Central Limit Theorem for a Gibbs Random Field
نویسنده
چکیده
We extend the validity of the implication of a local limit theorem from an integral one. Our extension eliminates the finite range assumption present in the previous works by using the cluster expansion to analyze the contribution from the tail of the potential. 1. Definitions and Results 1.1. Assumptions We consider a v-dimensional lattice spin system, to each χeZ is associated a spin sx that can take all the integer values lying between the two integers n,m. We denote by σ the max(| n , | m\ ), A configuration SΛ in a subset Λ, A c Z v is given by an element of {n, ...ra}, If Λ9M are two disjoint subsets of Z , we denote by SΛ v SM the spin configuration in Λ u M individuated by SΛ,SM. The interaction is given by a pair long range potential of the form J(x — y)sxsy where J is a real function on Z. We assume that: 1.2. Definition. Let A be a finite subset of Z. The Gibbs conditional distribution on the set of configurations in Λ, with condition szv\Λ is defined as: ί V } PΛ(SΛ\Z*\Λ) = P) X^A ( ~ 3^Λ + Σ M* V\J \/ZA(h,J)' (~ xφy xeΛ }
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تاریخ انتشار 2004