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

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.

39 pages, 1 figure