# F-KPP Scaling limit and selection principle for a Brunet-Derrida type particle system

## TL;DR

This paper analyzes a particle system with diffusion, branching, and selection, demonstrating its convergence to the F-KPP equation and showing it selects the minimal macroscopic speed as particle number grows.

## Contribution

It establishes the convergence of the particle system's empirical distribution to the F-KPP equation and proves the selection of minimal speed in the large particle limit.

## Key findings

- Empirical measure converges to F-KPP solution.
- System selects minimal macroscopic speed asymptotically.
- Provides a rigorous link between particle system and PDE.

## Abstract

We study a particle system with the following diffusion-branching-selection mechanism. Particles perform independent one dimensional Brownian motions and on top of that, at a constant rate, a pair of particles is chosen uniformly at random and both particles adopt the position of the rightmost one among them. We show that the cumulative distribution function of the empirical measure converges to a solution of the Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation and use this fact to prove that the system selects the minimal macroscopic speed as the number of particles goes to infinity.