Repeated bond traversal probabilities for the simple random walk

TL;DR

This paper analyzes the asymptotic behavior of bonds traversed exactly m times by a simple random walk in one and two dimensions, providing explicit scaling limits involving erfc and exponential functions.

Contribution

It explicitly determines the scaling limits of repeated bond traversal probabilities for simple random walks in low dimensions.

Findings

In dimension 1, the scaling limit involves the complementary error function (erfc).

In dimension 2, the scaling limit involves an exponential function.

Provides explicit formulas for the average number of bonds traversed exactly m times.

Abstract

We consider the average number B_m(t) of bonds traversed exactly m times by a t step simple random walk. We determine B_m(t) explicitly in the scaling limit t -> oo with m/sqrt(t) fixed in dimension d=1 and m/log(t) fixed in dimension d=2. The scaling function is an erfc in d=1 and an exponential in d=2.