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We study nearest-neighbors branching random walks started from a point at theinterior of a hypercube. We show that the probability that the process escapesthe hypercube is monotonically decreasing with respect to the distance of itsstarting point from the boundary. We derive as a consequence that at all timesthe number of particles at a site is monotonically decreasing with respect toits distance from the starting point.

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