BRW pressed against a hard wall

Abstract

We wish to find bounds on the order of the probability of a branching random walk on a d-ary tree being positive at all vertices. We also wish to find the expected value at a typical vertex under the condition that it is positive everywhere. The behavior that we are considering is that of entropic repulsion for Gaussian fields which is it’s behavior of drifting away when pressed against a hard wall. We show that the expected value is less than the expected maxima of BRW by a constant factor of \log n, which is a modification to the previous results on entropic repulsion of Gaussian fields.