Brownian paths as loop-decorated SLEs

TL;DR

This paper constructs a coupling between SLE$_2$ and Brownian motion by adding loops from a Brownian loop soup to a simple path, proving the law of a planar Brownian motion and resolving a conjecture of Lawler and Werner.

Contribution

It introduces a novel method to generate Brownian motion from SLE$_2$ using loop soups, confirming a longstanding conjecture and establishing a new coupling.

Findings

Adding loops from a Brownian loop soup to an SLE$_2$ path yields Brownian motion.

The constructed path law matches that of planar Brownian motion.

The method applies to off-critical setups like massive SLE$_2$.

Abstract

We construct an application, which takes as input a simple path and a possibly infinite collection of loops, and outputs a continuous path by adding the loops chronologically to the simple path as the simple path encounters them. By studying the regularity properties of this application and using lattice discretisations, we prove that chronologically adding the loops from a Brownian loop soup encountered by an independent radial SLE$_2$ path produces a continuous path which has the law of a planar Brownian motion. This resolves a conjecture of Lawler and Werner. This construction produces a coupling between SLE$_2$ and Brownian motion, and we further show that this joint law is the scaling limit of the loop-erased random walk and the random walk itself. The arguments are robust and can be applied for instance in the off-critical setup, where the scaling limit of loop-erased random walk is Makarov and Smirnov's massive SLE$_2$.