An example of Brunet-Derrida behavior for a branching-selection particle system on $\Z$

Jean Bérard (1)

1 ICJ - Institut Camille Jordan

We consider a branching-selection particle system on $\Z$ with $N \geq 1$ particles. During a branching step, each particle is replaced by two new particles, whose positions are shifted from that of the original particle by independently performing two random walk steps according to the distribution $p \delta_{1} + (1-p) \delta_{0}$, from the location of the original particle. During the selection step that follows, only the $N$ rightmost particles are kept among the $2N$ particles obtained at the branching step, to form a new population of $N$ particles. After a large number of iterated branching-selection steps, the displacement of the whole population of $N$ particles is ballistic, with deterministic asymptotic speed $v_{N}(p)$. As $N$ goes to infinity, $v_{N}(p)$ converges to a finite limit $v_{\infty}(p)$. The main result is that, for every $0

Domaines: Probabilités [math.PR]

Jean Bérard. An example of Brunet-Derrida behavior for a branching-selection particle system on $\Z$. 2008. ⟨hal-00335803v3⟩