Velocity of the $L$-branching Brownian motion

TL;DR

This paper analyzes the velocity growth of an $L$-branching Brownian motion system, proving it grows linearly with a specific asymptotic velocity as $L$ increases, confirming a conjecture in the physics literature.

Contribution

It provides a rigorous proof of the asymptotic behavior of the velocity $v_L$ in the $L$-branching Brownian motion model, connecting it with branching Brownian motion in a strip.

Findings

Velocity $v_L$ grows linearly with $L$

Asymptotic behavior of $v_L$ as $L o abla$ is $ o rac{ ext{constant}}{L^2}$

Confirmed the conjecture by Brunet, Derrida, Mueller, and Munier.

Abstract

We consider a branching-selection system of particles on the real line that evolves according to the following rules: each particle moves according to a Brownian motion during an exponential lifetime and then splits into two new particles and, when a particle is at a distance $L$ of the highest particle, it dies without splitting. This model has been introduced by Brunet, Derrida, Mueller and Munier in the physics literature and is called the $L$-branching Brownian motion. We show that the position of the system grows linearly at a velocity $v_L$ almost surely and we compute the asymptotic behavior of $v_L$ as $L$ tends to infinity: $v_L = \sqrt{2} - \pi^2 / 2 \sqrt{2} L^2 + o(1/L^2)$, as conjectured by Brunet, Derrida, Mueller and Munier. The proof makes use of results by Berestycki, Berestycki and Schweinsberg concerning branching Brownian motion in a strip.