Brunet-Derrida Behavior of Branching-Selection Particle Systems on the Line

Abstract

Abstract. We consider a class of branching-selection particle systems on R similar to the one considered by E. Brunet and B. Derrida in their 1997 paper "Shift in the velocity of a front due to a cutoff". Based on numerical simulations and heuristic arguments, Brunet and Derrida showed that, as the population size N of the particle system goes to infinity, the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate (log N ) −2 . In this paper, we give a rigorous mathematical proof of this fact, for the class of particle systems we consider. The proof makes use of ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and relies on a comparison of the particle system with a family of N independent branching random walks killed below a linear space-time barrier.

The genealogy of an exactly solvable Ornstein–Uhlenbeck type branching process with selection

“…Some of these conjectures have been verified for individual models. Bérard and Gouéré [1] proved that the corrections to the speed v − v N of many N -branching random walks are of the order (log N ) −2 , in accordance with the conjectures from [6]. These results haven been extended to branching random walks with different integrability conditions in [2,10,15], and to other related models in [7,14,17].…”

Fleming–Viot selects the minimal quasi-stationary distribution: The Galton–Watson case

“…Durrett and Remenik [14] establish propagation of chaos for a related continuous-space and time model, and then show that the limit of the empirical measure is characterized as the solution of a free-boundary integro-differential equation. Bérard and Gouéré [3] establish a conjecture of Brunet and Derrida for the speed of the rightmost particle for still a third microscopic model of F-KPP equation introduced in [7,8]. Maillard [21] obtains the precise behavior of the empirical measure of an approximation of the same model, building on the results of Berestycki, Berestycki and Schweinsberg [4], which establish the genealogy picture described in [7,8].…”

Two-sided infinite-bin models and analyticity for Barak–Erdős graphs

“…So far, two terms of the asymptotic behaviour of the speed of branching random walks with selection have been obtained by Bérard and Gouéré [3] for binary reproduction laws and extended in [16] to more general reproduction laws. In the special case of the infinite-bin model with uniform distribution, these results imply that…”