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

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.

Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length

“…In particular, such models can be designed in such a way that extreme value statistics interpolate between those of log-correlated variables and those of i.i.d. random variables [9,15,30]. The above conjecture provides a natural context, the Riemann zeta function, where such interpolated extreme value statistics would occur.…”