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

TL;DR

This paper proves that for branching random walks on a lattice, the chance of escaping a hypercube decreases as the starting point moves closer to the boundary, revealing monotonicity in escape probabilities and particle distribution.

Contribution

It establishes the monotonicity of escape probabilities and particle counts in branching random walks based on the initial position within a hypercube.

Findings

Escape probability decreases with proximity to the boundary.

Number of particles at a site decreases with distance from the start.

Monotonicity holds at all times during the process.

Abstract

We study nearest-neighbors branching random walks started from a point at the interior of a hypercube. We show that the probability that the process escapes the hypercube is monotonically decreasing with respect to the distance of its starting point from the boundary. 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 point.