Branching capacity of a random walk range

Abstract

We consider the branching capacity of the range of a simple random walk on $\mathbb Z^d$, with $d \ge 5$, and show that it falls in the same universality class as the volume and the capacity of the range of simple random walks and branching random walks. To be more precise we prove a law of large numbers in dimension $d \ge 6$, with a logarithmic correction in dimension 6, and identify the correct order of growth in dimension 5. The main original part is the law of large numbers in dimension 6, for which one needs a precise asymptotic of the non-intersection probability of an infinite invariant critical tree-indexed walk with a two-sided simple random walk. The result is analogous to the estimate proved by Lawler for the non-intersection probability of an infinite random walk with a two-sided walk in dimension four. While the general strategy of Lawler's proof still applies in this new setting, many steps require new ingredients.