On Brownian motion, simple paths, and loops

TL;DR

This paper introduces a decomposition of Brownian motion into a simple path and a loop soup, providing new insights into the structure of scaling limits of loop erased random walks, especially in three dimensions.

Contribution

It proves that subsequential scaling limits of loop erased random walk are simple paths in three dimensions, and establishes a Hausdorff dimension bound for these limits.

Findings

Subsequential scaling limits are simple paths in 3D.

Hausdorff dimension of limits lies in (1, 5/3].

Decomposition characterizes Brownian motion into a simple path and loop soup.

Abstract

We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies in $(1,\frac53]$. We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results would give a new strategy to the existence of the scaling limit of the loop erased random walk and its rotational invariance.