  In this note 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
behaviour. In a first version of the model the mean field limit reproduces the
behaviour of the original REM, while a second version of the model exhibits a
first order phase transition with a finite latent heat.
