Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve

TL;DR

This paper proves that the loop-erased random walk on a planar graph converges to the chordal SLE(2) curve in the scaling limit, under certain invariance principles, linking discrete models to continuous conformally invariant curves.

Contribution

It establishes the convergence of loop-erased random walks on planar graphs to chordal SLE(2), extending the understanding of scaling limits in planar statistical physics models.

Findings

Loop-erased random walk converges to SLE(2) in the scaling limit.

Convergence holds under invariance principles for both the walk and its dual.

Provides rigorous connection between discrete random walks and continuous SLE curves.

Abstract

In this paper we consider the natural random walk on a planar graph and scale it by a small positive number $\delta$. Given a simply connected domain $D$ and its two boundary points $a$ and $b$, we start the scaled walk at a vertex of the graph nearby $a$ and condition it on its exiting $D$ through a vertex nearby $b$, and prove that the loop erasure of the conditioned walk converges, as $\delta \downarrow 0$, to the chordal SLE$_{2}$ that connects $a$ and $b$ in $D$, provided that an invariance principle is valid for both the random walk and the dual walk of it.