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 $\alpha \in \left[0,1\right]$: choosing $\alpha=0$ yields the random energy model by Derrida (REM), whereas $\alpha=1$ corresponds to the branching random walk (BRW). When the parameter $\alpha$ increases, the level of the maximum of the field decreases smoothly from the REM- to the BRW-value. However, as long as $\alpha<1$ strictly, the limiting extremal process is always Poissonian.