Finite-size corrections to the speed of a branching-selection process

TL;DR

This paper analyzes a particle system with branching and selection, focusing on finite-size effects on the speed of the traveling front, especially in the Weibull domain of attraction, revealing universal correction features.

Contribution

It provides a detailed analysis of finite-size corrections to the front speed in a branching-selection process within the Weibull domain, highlighting universal behaviors.

Findings

Finite-size correction to speed depends on tail probabilities.

Universal features of correction are identified in Weibull domain.

Model analysis as population size N tends to infinity.

Abstract

We consider a particle system studied by E. Brunet and B. Derrida, which evolves according to a branching mechanism with selection of the fittest keeping the population size fixed and equal to $N$. The particles remain grouped and move like a travelling front driven by a random noise with a deterministic speed. Because of its mean-field structure, the model can be further analysed as $N \to \infty$. We focus on the case where the noise lies in the max-domain of attraction of the Weibull extreme value distribution and show that under mild conditions the correction to the speed has universal features depending on the tail probabilities.