An upper bound for the probability of visiting a distant point by critical branching random walk in $\mathbb{Z}^4$

TL;DR

This paper establishes an upper bound on the probability that a critical branching random walk in four-dimensional integer lattice visits a distant point, showing it decreases proportionally to 1 over the squared distance times a logarithmic factor.

Contribution

It provides a precise upper bound for the visiting probability of distant points by critical branching random walks in 4, advancing understanding of spatial behavior in such processes.

Findings

Probability bound is proportional to 1/(|a|^2 log |a|)

Visits to distant points become increasingly unlikely as distance grows

Results are specific to critical branching random walks in 4

Abstract

In this paper, we study the probability of visiting a distant point $a\in \mathbb{Z}^4$ by critical branching random walk starting from the origin. We prove that this probability is bounded by $1/(|a|^2\log |a|)$ up to a constant.