Hausdorff dimension of the scaling limit of loop-erased random walk in three dimensions

TL;DR

This paper proves that the Hausdorff dimension of the scaling limit of 3D loop-erased random walk equals the expected growth exponent of its length, establishing a precise fractal dimension in three dimensions.

Contribution

It establishes that the Hausdorff dimension of the 3D loop-erased random walk's scaling limit equals the expected length exponent, linking geometric and probabilistic properties.

Findings

Hausdorff dimension equals the exponent β almost surely

Expected length of loop-erased walk scales as n^β

Dimension matches the growth rate exponent

Abstract

Let $M_{n}$ be the length (number of steps) of the loop-erasure of a simple random walk up to the first exit from a ball of radius $n$ centered at its starting point. It is shown in [18] that there exists $\beta \in (1, \frac{5}{3}]$ such that $E (M_{n} )$ is of order $n^{\beta}$ in 3 dimensions. In the present article, we show that the Hausdorff dimension of the scaling limit of the loop-erased random walk in 3 dimensions is equal to $\beta$ almost surely.