arXiv subjects

Mohamed Ali Belloum

Publications and source records attributed to Mohamed Ali Belloum.

3 recordsLinked to original sources

arXiv2026

The extremal process of a cascading family of branching Brownian motion

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 $2$.

math.PR↗

arXiv2021

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

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 $GREM(N^α)$, $α\in{[0,1)}$ 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 decorate Poisson point process by Madaule. In this paper we propose a generalized model of the $GREM(N^α)$, 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.

math.PR↗

arXiv2021

Anomalous spreading in reducible multitype branching Brownian motion

We consider a two-type reducible branching Brownian motion, defined as a particle system on the real line in which particles of two types move according to independent Brownian motion and create offspring at constant rate. Particles of type $1$ can give birth to particles of types $1$ and $2$, but particles of type $2$ only give birth to descendants of type $2$. Under some specific conditions, Biggins (2012) shows that this process exhibit an anomalous spreading behaviour: the rightmost particle at time $t$ is much further than the expected position for the rightmost particle in a branching Brownian motion consisting only of particles of type $1$ or of type $2$. This anomalous spreading also has been investigated from a reaction-diffusion equation standpoint by Holzer (2014,2016). The aim of this article is to study the asymptotic behaviour of the position of the furthest particles in the two-type reducible branching Brownian motion, obtaining in particular tight estimates for the median of the maximal displacement.

math.PR↗