On the existence of a glass transition in a Random Energy Model

We consider a generalized version of the Random Energy Model in which the energy of each configuration is given by the sum of $N$ independent contributions ("local energies") with finite variances but otherwise arbitrary statistics. Using the large deviation formalism, we find that the glass transition generically exists when local energies have a smooth distribution. In contrast, if the distribution of the local energies has a {Dirac mass} at the minimal energy (e.g., if local energies take discrete values), the glass transition ceases to exist if the number of energy levels grows sufficiently fast with system size. This shows that statistical independence of energy levels does not imply the existence of a glass transition.