Monotonicity of escape probabilities for branching random walks on $\Z^{d}$

Achillefs Tzioufas

Published 2019-11-21Version 1

We study nearest-neighbors branching random walks started by a single particle at the interior of a hypercube. We show that the probability of its progeny escaping a hypercube is monotonically decreasing with respect to the distance of the starting point from the boundary of the hypercube. We derive as a consequence that at all times the number of particles at a site is monotonically decreasing with respect to its distance from the starting site.