Branching random walk subject to a hard wall constraint

We consider a Gaussian branching random walk under the constraint that the values at all leaves at generation n are non-negative. We obtain a remarkably precise description of the conditional law and the conditional field. The conditioning leads to an upward shift of the whole field. We obtain sharp estimates on this upward shift (up to o(1) terms). We show that the properly rescaled maximum converges to a Gumbel distribution (without a random shift!), and the rescaled minimum is exponentially distributed. We use tools from DGFFs on general graphs and estimates on random walks that are weakly attracted to zero either through a pinning potential or a drift. The talk is based on joint work with M. Fels (Technion) and O. Louidor(Technion).