Crossover behaviour of a one dimensional Random Energy Model
  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. 
Matteo Campellone, Silvio Franz and Giorgio Parisi