Four-dimensional loop-erased random walk

Abstract

The loop-erased random walk (LERW) in $\mathbb{Z}^{4}$ is the process obtained by erasing loops chronologically for a simple random walk. We prove that the escape probability of the LERW renormalized by $(\log n)^{\frac{1}{3}}$ converges almost surely and in $L^{p}$ for all $p>0$. Along the way, we extend previous results by the first author building on slowly recurrent sets. We provide two applications for the escape probability. We construct the two-sided LERW, and we construct a $\pm 1$ spin model coupled with the wired spanning forests on $\mathbb{Z}^{4}$ with the bi-Laplacian Gaussian field on $\mathbb{R}^{4}$ as its scaling limit.