Random walk on the range of random walk

TL;DR

This paper investigates the behavior of a random walk on the range of another random walk in high dimensions, establishing different scaling limits and corrections depending on the dimension, with precise results for dimensions five and above.

Contribution

It provides the first detailed analysis of the scaling limits of a random walk on the range of a simple random walk across different dimensions, highlighting the importance of logarithmic corrections in four dimensions.

Findings

For d ≥ 5, established quenched and annealed scaling limits showing intersections are negligible.

For d=4, identified the necessity of logarithmic corrections in scaling and return probability.

Demonstrated that in four dimensions, asymptotic behavior requires logarithmic adjustments.

Abstract

We study the random walk $X$ on the range of a simple random walk on $\mathbb{Z}^d$ in dimensions $d\geq 4$. When $d\geq 5$ we establish quenched and annealed scaling limits for the process $X$, which show that the intersections of the original simple random walk path are essentially unimportant. For $d=4$ our results are less precise, but we are able to show that any scaling limit for $X$ will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when $d=4$ similar logarithmic corrections are necessary in describing the asymptotic behaviour of the return probability of $X$ to the origin.