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We study a generalization of the model introduced by Kistler and Schmidt in$2015$, that interpolates between the random energy model (REM) and thebranching random walk (BRW). More precisely, we are interested in theasymptotic behaviour of the extremal process associated to this model. Kistlerand Schmidt show that the extremal process of the $GREM(N^{\alpha})$,$\alpha\in{[0,1)}$ converges weakly to a simple Poisson point process. Thiscontrasts 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 $GREM(N^{\alpha})$, thathas the structure of a tree with $k_n$ levels, where $(k_n\leq n)$ is anon-decreasing sequence of positive integers. We study a generalized case,where the position of the particles are not necessarily Gaussian variables andthe 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 thenon Gaussian case) the decoration disappears and we have convergence to asimple Poisson point process.

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