Large and moderate deviations in the Ising model

Abstract

The theory of large and moderate deviations of sums of random variables is now a wide and fastly growing branch of probability theory (see references in §1.9). It was created initially in the framework of the theory of sums of independent identically distributed random variables and then extended to a wide class of random processes, i.e., random functions in one variable, with some general conditions of weak dependence traditional for the theory of random processes. It was found that in these cases the asymptotic structure of probabilities of deviations is completely similar to the situation for the case of independent variables. We shall call this case a classical behavior of large and moderate deviations. In the last few years probability theory has turned to the study of random fields, i.e., random functions in several variables. It became clear that the main difference of this multidimensional case is the possibility of a phase transition, which occurs even in the simplest typical situations. It turns out that the existence of a phase transition can radically change the asymptotic behavior of large and moderate deviations. The aim of this paper is to study these new effects and to compare them with the classical behavior. It turns out also that this behavior is closely related to the physical phenomenon of condensation of droplets of one phase inside another phase. To simplify the exposition we will present the results for the simplest nontrivial Gibbs field, namely the ferromagnetic Ising model. Possible generalizations are discussed in §1.9.