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

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 $0<p<1/2$, as $N$ goes to infinity, the order of magnitude of the difference $v_{\infty}(p)- v_{N}(p)$ is $\log(N)^{-2}$. This is called Brunet-Derrida behavior in reference to the 1997 paper by E. Brunet and B. Derrida "Shift in the velocity of a front due to a cutoff" (see the reference within the paper), where such a behavior is established for a similar branching-selection particle system, using both numerical simulations and heuristic arguments.