$N$-Branching random walk with $\alpha$-stable spine

TL;DR

This paper investigates a branching-selection particle system with a fixed population size, focusing on the behavior of the critical spine modeled as an $oldsymbol{ ext{alpha}}$-stable random walk, bridging exponential and heavy tail displacement cases.

Contribution

It introduces an intermediate model where the critical spine follows an $oldsymbol{ ext{alpha}}$-stable distribution, extending previous work on exponential and heavy tail displacement regimes.

Findings

Analysis of the speed of the particle cloud drift.

Characterization of the critical spine as an $oldsymbol{ ext{alpha}}$-stable process.

Insights into the transition between exponential and heavy tail behaviors.

Abstract

We consider a branching-selection particle system on the real line, introduced by Brunet and Derrida. In this model the size of the population is fixed to a constant $N$. At each step individuals in the population reproduce independently, making children around their current position. Only the $N$ rightmost children survive to reproduce at the next step. B\'erard and Gou\'er\'e studied the speed at which the cloud of individuals drifts, assuming the tails of the displacement decays at exponential rate; B\'erard and Maillard took interest in the case of heavy tail displacements. We take interest in an intermediate model, considering branching random walks in which the critical spine behaves as an $\alpha$-stable random walk.