Branching capacity of a random walk in $\mathbb Z^5$

TL;DR

This paper investigates the branching capacity of a random walk in five-dimensional integer lattice, revealing a convergence to Brownian snake capacity and analyzing intersection probabilities of branching and regular random walks.

Contribution

It extends understanding of branching capacity in five dimensions, showing convergence to Brownian snake capacity and analyzing critical intersection probabilities.

Findings

Renormalized branching capacity converges to Brownian snake capacity.

Established intersection probability estimates between branching and simple random walks.

Identified a critical transition at dimension d=6.

Abstract

We are interested in the branching capacity of the range of a random walk in $\mathbb Z^d$.Schapira [28] has recently obtained precise asymptotics in the case $d\ge 6$ and has demonstrated a transition at dimension $d=6$. We study the case $d=5$ and prove that the renormalized branching capacity converges in law to the Brownian snake capacity of the range of a Brownian motion. The main step in the proof relies on studying the intersection probability between the range of a critical Branching random walk and that of a random walk, which is of independent interest.