The branching random walk subject to a hard-wall constraint

We study the branching random walk under a "hard wall constraint", namely when the heights of all particles in the most recent generation are conditioned to be positive. We obtain sharp asymptotics for the probability of this event and for various statistics, conditional on its occurrence. In particular, we identify the repulsion profile followed by the conditional field, and derive limits in law for its maximum, minimum and associated additive martingales. These results show, among other things, that the laws of the conditional and unconditional fields are mutually singular in the limit.

This is joint work with Lisa Hartung (University of Mainz) and Maximilian Fels (Technion).