Brunet-Derrida particle systems, free boundary problems and Weiner-Hopf equations
Rick Durrett and Daniel Remenik
Abstract.
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 tends to infinity, the tail
distribution of the empirical measure associated to the system converges to the solution
of a free boundary integro-differential 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 and 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 are solutions of Weiner-Hopf equations.
Preprint Ann. Prob., to appear
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