Crossover behavior of a one-dimensional random energy model

In this paper we formulate a finite-dimensional generalization of the random energy model (REM) where we introduce a geometry and spatial correlations between energies. We study the model in dimension one by transfer matrix techniques and we look at the crossover from one-dimensional to mean-field behavior. In a first version of the model the mean field limit reproduces the behavior of the original REM, while a second version of the model exhibits a first-order phase transition with a finite latent heat, [S1063-651X(98)00211-6].