Branching random walk pressed against a hard wall

Abstract

We consider the Branching Random Walk on d-ary tree of height n under the presence of a hard wall which restricts each value to be positive. The question of behavior of Gaussian process with long range interactions under the presence of a hard wall has been addressed, and a relation with the expected maxima of the processes has been established. We find the probability of event corresponding to the hard wall and show that the conditional expectation of the gaussian variable at a typical vertex, under positivity, is at least less than the expected maxima by order of log n.