From derrida’s random energy model to branching random walks: From 1 to 3

Abstract

We study the extremes of a class of Gaussian fields with in-built hierarchical structure. The number of scales in the underlying trees depends on a parameter α Є [0, 1]: choosing α = 0 yields the random energy model by Derrida (REM), whereas α = 1 corresponds to the branching random walk (BRW). When the parameter α increases, the level of the maximum of the field decreases smoothly from the REM- to the BRWvalue. However, as long as α < 1 strictly, the limiting extremal process is always Poissonian.