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

TL;DR

This paper derives the scaling limit of the loop-erased random walk branch in uniform spanning trees within topological polygons, revealing new variants of SLE$_2$ for multiple boundary points.

Contribution

It generalizes previous results by establishing the scaling limit of LERW branches in UST for polygons with more than two boundary points, showing they are not in the SLE$_2( ho)$ family for N≥3.

Findings

Scaling limit of LERW branch is a variant of SLE$_2$.

For N=2, the limit is SLE$_2(-1,-1;-1,-1).

For N≥3, the limit is not in the SLE$_2( ho)$ family.

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 $N\ge 3$.