Brunet-Derrida behavior of branching-selection particle systems on the line

TL;DR

This paper rigorously proves that the velocity of certain branching-selection particle systems on the line converges to a limit at a rate proportional to the inverse square of the logarithm of the population size, confirming earlier heuristic predictions.

Contribution

It provides a rigorous mathematical proof of the Brunet-Derrida velocity convergence rate for a class of particle systems, extending previous heuristic and numerical findings.

Findings

Velocity converges to a limit at rate (log N)^{-2}

Comparison with branching random walks is effective for analysis

Confirms Brunet-Derrida's heuristic predictions

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.