A generalized model interpolating between the random energy model and the branching random walk

Abstract

We study a generalization of the model introduced by Kistler and Schmidt in $2015$, that interpolates between the random energy model (REM) and the branching random walk (BRW). More precisely, we are interested in the asymptotic behaviour of the extremal process associated to this model. Kistler and Schmidt show that the extremal process of the $GREM(N^{\alpha})$, $\alpha\in{[0,1)}$ converges weakly to a simple Poisson point process. This contrasts with the extremal process of the branching random walk $(\alpha=1)$ which was shown to converge toward a decorate Poisson point process by Madaule. In this paper we propose a generalized model of the $GREM(N^{\alpha})$, that has the structure of a tree with $k_n$ levels, where $(k_n\leq n)$ is a non-decreasing sequence of positive integers. We study a generalized case, where the position of the particles are not necessarily Gaussian variables and the reproduction law is not necessarily binary. We show that as long as $b_n=\lfloor{\frac{n}{k_n}}\rfloor\to_{n\to \infty}\infty$ in the Gaussian case (with the assumption $\frac{b_n}{\log(n)^2}\to\infty$ as $n\to \infty$ in the non Gaussian case) the decoration disappears and we have convergence to a simple Poisson point process.