Logarithmic correction to resistance

TL;DR

This paper investigates the electrical resistance in a critical branching random walk in six dimensions, revealing a logarithmic correction that refines previous resistance estimates.

Contribution

It establishes a universal logarithmic correction to resistance in the incipient infinite oriented branching random walk in six dimensions.

Findings

Resistance between root and level n is O(n log^{-ξ} n)

The correction exponent ξ is independent of model specifics

Results apply under certain moment assumptions

Abstract

We study the trace of the incipient infinite oriented branching random walk in $\mathbb{Z}^d \times \mathbb{Z}_+$ when the dimension is $d = 6$. Under suitable moment assumptions, we show that the electrical resistance between the root and level $n$ is $O(n \log^{-\xi}n )$ for a $\xi > 0$ that does not depend on details of the model.