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

Qingsan Zhu

doi:10.1214/19-ecp228

Abstract

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

Highlights

A branching random walk is a discrete-time particle system in Zd as the following
We study the probability of visiting a distant point a ∈ Z4 by a critical branching random walk starting at the origin
We prove that this probability is bounded by 1/(|a|2 log |a|) up to a constant factor

Summary

Introduction

A branching random walk is a discrete-time particle system in Zd as the following. Fix a probability measure μ on N, called offspring distribution, and another probability measure θ on Zd, called jump distribution. We study the probability of visiting a distant point a ∈ Z4 by a critical branching random walk starting at the origin. A branching random walk is a discrete-time particle system in Zd as the following.

Results

Conclusion