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We consider a general model of discrete-time random walk X-t on the lattice (nu), nu = 1,..., in a random environment xi={xi(t,x):(t,x)is an element of(nu+1)} with i.i.d. components xi(t,x). Previous results on the a.s. validity of the Central Limit Theorem for the quenched model required a small stochasticity condition. In this paper we show that the result holds provided only that an obvious non-degeneracy condition is met. The proof is based on the analysis of a suitable generating function, which allows to estimate L-2 norms by contour integrals

A system of random variables (spins) Sx, x~2U, taking on values in IR is considered. Conditional probabilities for the joint distributions of a finite number of spins are prescribed; a DLR measure is then a process onthe 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 JxySx Sy, Jxy~lR and via a self-interaction F(Sx), which, as ISx[--,oo, diverges at least quadratically [3]. By use of a technique introduced inis 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 ~ as limits of Gibbs measures for finitely many spins in a wide class of boundary conditions, Theorem 1.2. Uniqueness in ~ is then discussed by means of a theorem by Dobrugin [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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