Loop-Erased Random Walk Branch of Uniform Spanning Tree in Topological Polygons

Abstract

We consider uniform spanning tree (UST) in topological polygons with $2N$ marked points on the boundary with alternating boundary conditions. In [LPW21], the authors derive the scaling limit of the Peano curve in the UST. They are variants of SLE${}_8$. In this article, we derive the scaling limit of the loop-erased random walk branch (LERW) in the UST. They are variants of SLE${}_2$. The conclusion is a generalization of [HLW20,Theorem 1.6] where the authors derive the scaling limit of the LERW branch of UST when $N=2$. When $N=2$, the limiting law is SLE${}_2(-1,-1;-1,-1)$. However, the limiting law is nolonger in the family of SLE${}_2(\rho )$ process as long as $N\ge 3$.

18 pages, 1 figure