Limit theorems for sums of dependent random variables occurring in statistical mechanics

We study the asymptotic behavior of partial sums S_nfor certain triangular arrays of dependent, identically distributed random variables which arise naturally in statistical mechanics. A typical result is that under appropriate assumptions there exist a real number m, a positive real number λ, and a positive integer k so that (S_n-nm)/n^1-1/2k converges weakly to a random variable with density proportional to exp(-λ|s|^2k/(2k)!). We explain the relation of these results to topics in Gaussian quadrature, to the theory of moment spaces, and to critical phenomena in physical systems.