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

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.

24 pages