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

@article{Tzioufas2019MonotonicityOE,
  title={Monotonicity of escape probabilities for branching random walks on \$\Z^\{d\}\$},
  author={Achillefs Tzioufas},
  journal={arXiv: Probability},
  year={2019}
}
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. 

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