Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations

We consider a branching-selection system in R with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N → ∞, the empirical measure process associated to the system converges in distribution to a deterministic measure-valued process whose densities solve a free boundary integrodifferential equation. We also show that this equation has a unique traveling wave solution traveling at speed c or no such solution depending on whether c > a or c ≤ a, where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of Wiener-Hopf equations.