The Alloy Model: Phase Transitions and Diagrams for a Random Energy Model with Mixtures

In this paper we introduce the alloy model, which is a variant of Derrida's random energy model (REM). The alloy model assumes that energy levels are independent and identically distributed (iid) random variables, whose distribution is a mixture of two distribution from the same location-scale family. These families are assumed to have either Weibull or Gumbel tails. Particular attention is paid to the case of normal distributions. For these we get more explicit results, which show that, for certain choices of the parameters, we can have two phase transitions, one first order and one second order.