Large Deviations and Random Energy Models

A unified treatment for the existence of free energy in several random energy models is presented. If the sequence of distributions associated with the particle systems obeys a large deviation principle, then the free energy exists almost surely. This includes all the known cases as well as some heavy-tailed distributions. 1. Introduction. The purpose of this note is to bring out the essence involved in the existence theorems for the limiting free energy in several random energy models. Basically, they are corollaries to the large deviation principle (LDP) obeyed by certain empirical measures coupled with Varad-han's lemma. Since this note is addressed to probabilists, the results are first formulated in the large deviation setting and then the consequences to the spin glass models are explained in each section. At the first sight it may appear that the large deviation principles proved here are nothing but the well known principles for empirical measures, but however a little reflection shows that it is not so. For large deviation terminology, we refer to [4, 9]. We could have given the most general result of section 5 and then deduced all the results from it. Instead, we decided to go from simple model to the more general one, so that it is easier for the reader to follow.