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 an earlier work by Liu-Peltola-Wu, the authors derive the scaling limit of the Peano curve in the UST. They are variants of SLE8. In this article, we derive the scaling limit of the loop-erased random walk branch (LERW) in the UST. They are variants of SLE2. The conclusion is a generalization of an earlier work by Han-Liu-Wu where the authors derive the scaling limit of the LERW branch of UST when N=2. When N=2, the limiting law is SLE2(−1,−1;−1,−1). However, the limiting law is no longer in the family of SLE2(ρ) process as long as N≥3.