Graph distance and effective resistance of the four-dimensional random walk trace
Daisuke Shiraishi, Satomi Watanabe
TL;DR
This work resolves the four-dimensional critical case for the trace of a simple random walk by deriving sharp asymptotics for the graph distance and effective resistance between the origin and the endpoint of the trace. The authors develop a two-scale decomposition and leverage refined long-range intersection estimates to obtain a precise logarithmic correction, showing both $E(D_n)$ and $E(R_n)$ scale as $n(\log n)^{-1/2}$ with explicit constants and a polynomially small error term. The central methodological thrust is to analyze the slowly varying function $\psi(n)=E(D_n)/n$ via the relation between $\psi(n)$ and $\psi(2n)$, facilitated by a detailed study of intersections of long segments of the walk and multi-walk non-intersections, encapsulated in new propositions and lemmas. These results not only settle a conjectured logarithmic correction at criticality but also enable refined fluctuation analyses and potential applications to heat kernel behavior on the random trace, as well as to the study of collisions of independent walks on the trace graph.
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.