Four dimensional loop-erased random walk

TL;DR

This paper studies the four-dimensional loop-erased random walk (LERW), proving the almost sure convergence of its escape probability scaled by a logarithmic factor, and introduces applications including a two-sided LERW and a spin model linked to the bi-Laplacian Gaussian field.

Contribution

The paper extends previous results on LERW in four dimensions by establishing convergence of escape probabilities and constructing new models like the two-sided LERW and a coupled spin model with a Gaussian field.

Findings

Escape probability scaled by $( ext{log } n)^{1/3}$ converges almost surely.

Construction of the two-sided LERW.

Development of a spin model coupled with spanning forests and the bi-Laplacian Gaussian field.

Abstract

The loop-erased random walk (LERW) in $\mathbb{Z}^4$ is the process obtained by erasing loops chronologically for 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.