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We consider a branching-selection system in $\mathbb {R}$ with $N$ particleswhich give birth independently at rate 1 and where after each birth theleftmost particle is erased, keeping the number of particles constant. We showthat, as $N\to\infty$, the empirical measure process associated to the systemconverges in distribution to a deterministic measure-valued process whosedensities solve a free boundary integro-differential equation. We also showthat this equation has a unique traveling wave solution traveling at speed $c$or no such solution depending on whether $c\geq a$ or $c<a$, where $a$ is theasymptotic speed of the branching random walk obtained by ignoring the removalof the leftmost particles in our process. The traveling wave solutionscorrespond to solutions of Wiener-Hopf equations.

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