Derrida’s random energy models: From spin glasses to the extremes of correlated random fields

Abstract

We discuss Derrida’s random energy models under the light of the recent advances in the study of the extremes of highly correlated random fields. In particular, we present a refinement of the second moment method which provides a unifying approach to models where multiple scales can be identified, such is the case for e.g. branching diffusions, the 2-dim Gaussian free field, certain issues of percolation in high dimensions, or cover times. The method identifies some universal mechanisms which seemingly play a fundamental role also in the behavior of the extremes of the characteristic polynomials of certain random matrix ensembles, or in the extremes of the Riemann ζ-function along the critical line.