Last passage percolation and traveling fronts

TL;DR

This paper studies a particle system modeling last passage percolation with stochastic dynamics, analyzing the front's speed and profile as the system size grows, revealing different behaviors depending on the noise distribution.

Contribution

It provides new estimates for the front speed and profile in a mean-field last passage percolation model, highlighting the role of Gumbel distribution and Lévy processes in the scaling limit.

Findings

Gumbel distribution governs particle jumps in the model.

Scaling limit is a Lévy process under Gumbel noise.

Bounded jumps lead to different asymptotic behavior.

Abstract

We consider a system of N particles with a stochastic dynamics introduced by Brunet and Derrida. The particles can be interpreted as last passage times in directed percolation on {1,...,N} of mean-field type. The particles remain grouped and move like a traveling wave, subject to discretization and driven by a random noise. As N increases, we obtain estimates for the speed of the front and its profile, for different laws of the driving noise. The Gumbel distribution plays a central role for the particle jumps, and we show that the scaling limit is a L\'evy process in this case. The case of bounded jumps yields a completely different behavior.