Electrical resistance of the low dimensional critical branching random walk

TL;DR

This paper investigates the electrical resistance in low-dimensional critical branching random walks, establishing bounds that answer a previously open question in the field.

Contribution

It provides a universal bound on electrical resistance in low-dimensional critical branching random walks, addressing an open problem in the literature.

Findings

Electrical resistance between origin and generation n is bounded by O(n^{1-alpha}) for some alpha>0.

The result applies to dimensions d<6.

It confirms a conjecture posed by Barlow, Járai, Kumagai, and Slade.

Abstract

We show that the electrical resistance between the origin and generation n of the incipient infinite oriented branching random walk in dimensions d<6 is O(n^{1-alpha}) for some universal constant alpha>0. This answers a question of Barlow, J\'arai, Kumagai and Slade [2].