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

2021/08/24 by Mingchang Liu, Hao Wu, Liu, Mingchang +1 · 1 citation
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2108.10500

openalex publication_date 2021/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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 [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 SLE₂(-1,-1; -1, -1). However, the limiting law is nolonger in the family of SLE₂(ρ) process as long as N≥ 3.

Citations

• Hypergeometric SLE with κ=8: Convergence of UST and LERW in Topological Rectangles
• Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in c=-2 Logarithmic CFT

Cited by

• Splitting algorithm and normed convergence for drawing the random Loewner curves

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