Branching random walks and Minkowski sum of random walks

TL;DR

This paper demonstrates the intersection-equivalence of the range of critical branching random walks and Minkowski sums of simple random walk ranges in high dimensions, extending classical results on Wiener sausages and linking capacity with branching capacity.

Contribution

It establishes intersection-equivalence in high dimensions and extends a law of large numbers for Minkowski sums of random walk ranges, connecting capacity and branching capacity.

Findings

Range and Minkowski sums hit finite sets with comparable probability

Volume of Minkowski sausage converges to capacity-related quantity

New relation between capacity and branching capacity

Abstract

We show that the range of a critical branching random walk conditioned to survive forever and the Minkowski sum of two independent simple random walk ranges are intersection-equivalent in any dimension $d\ge 5$, in the sense that they hit any finite set with comparable probability, as their common starting point is sufficiently far away from the set to be hit. Furthermore, we extend a discrete version of Kesten, Spitzer and Whitman's result on the law of large numbers for the volume of a Wiener sausage. Here, the sausage is made of the Minkowski sum of $N$ independent simple random walk ranges in $\mathbb{Z}^d$, with $d>2N$, and of a finite set $A\subset \mathbb{Z}^d$. When properly normalised the volume of the sausage converges to a quantity equivalent to the capacity of $A$ with respect to the kernel $K(x,y)=(1+\|x-y\|)^{2N-d}$. As a consequence, we establish a new relation between capacity and {\it branching capacity}.