No abstract

We study a generalization of the model introduced in [22] 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. In [22], Kistler and Schmidt show that the extremal process of the GREM (N α ), α ∈ [0, 1) converges weakly to a simple Poisson point process. This contrasts with the extremal process of the branching random walk (α = 1) which was shown to converge toward a decorated Poisson point process by Madaule [20]. In this paper we propose a generalized model of the GREM (N α ), that has the structure of a tree with kn levels, where (kn ≤ n) is a non-decreasing sequence of positive integers. We show that as long as kn n →n→∞ 0, the decoration disappears and we have convergence to a simple Poisson point process. 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 study the asymptotic behaviour of the extremal process of a cascading family of branching Brownian motions. This is a particle system on the real line such that each particle has a type in addition to his position. Particles of type 1 move on the real line according to Brownian motions and branch at rate 1 into two children of type 1. Furthermore, at rate α, they give birth to children too of type 2. Particles of type 2 move according to standard Brownian motion and branch at rate 1, but cannot give birth to descendants of type 1. We obtain the asymptotic behaviour of the extremal process of particles of type 2.