Time spent in a ball by a critical branching random walk

TL;DR

This paper investigates the duration a critical branching random walk spends within a ball in high dimensions, revealing unique features of the critical dimension four and analyzing boundary transport phenomena.

Contribution

It provides new insights into the tail behavior of time spent in a ball for critical branching random walks, especially in dimension four, building on recent related results.

Findings

Characterizes the tail distribution of time spent in a ball in high dimensions

Highlights unique properties of the critical dimension four

Analyzes the number of walks transported on the boundary of distant balls

Abstract

We study a critical branching random walk on $\mathbb Z^d$. We focus on the tail of the time spent in a ball, and our study, in dimension four and higher, sheds new light on the recent result of Angel, Hutchcroft and Jarai, in particular on the special features of the critical dimension four. Finally, we analyse the number of walks transported by the branching random walk on the boundary of a distant ball.