Last Passage Percolation and Traveling Fronts.

We consider a system of N particles with a stochastic dynamics introduced by Brunet and Derrida (Phys. Rev. E 70:016106, ). 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 front, 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. As shown in Brunet and Derrida (Phys. Rev. E 70:016106, ), the model with Gumbel distributed jumps has a simple structure. We establish that the scaling limit is a Lévy process in this case. We study other jump distributions. We prove a result showing that the limit for large N is stable under small perturbations of the Gumbel. In the opposite case of bounded jumps, a completely different behavior is found, where finite-size corrections are extremely small. [ABSTRACT FROM AUTHOR]