Graph distance and effective resistance of the four-dimensional random walk trace

TL;DR

This paper provides precise asymptotic estimates for the expected graph distance and resistance metric between the start and end points of a four-dimensional random walk trace, refining prior results.

Contribution

It introduces sharper asymptotic estimates for graph distance and resistance in four-dimensional random walk traces, improving understanding of their geometric properties.

Findings

Asymptotic estimate for expected graph distance

Asymptotic estimate for resistance metric

Refinement of previous results

Abstract

Refining previous results, we establish a sharp asymptotic estimate on the expected graph distance between the origin and the terminal point of the trace of the first $n$ steps of the walk. A similar conclusion is drawn for the resistance metric.