Alessandro Pellegrinotti

Existence and uniqueness of DLR measures for unbounded spin systems

SummaryA 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

\nu

\mathfrak{E} = \{ v is DLR|\exists c(v), \mathop {sup}\limits_{x \in \mathbb{Z}^v } \int {v(dS_x )} |S_x | < c(v)\}

Convergence to stationary states for infinite harmonic systems

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

Ergodic Systems of N Balls in a Billiard Table

\mathfrak{E}

Ergodic properties of a semi-infinite one-dimensional system of statistical mechanics

as limits of Gibbs measures for finitely many spins in a wide class of boundary conditions, Theorem 1.2. Uniqueness in