The Loewner difference equation and convergence of loop-erased random walk

2016/11/04 by Gregory F. Lawler, Lawler, Gregory F., Fredrik Viklund +1
Mathematics · Physics and Astronomy · #Complex Variables (math.CV) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1611.01406

openalex publication_date 2016/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We revisit the convergence of loop-erased random walk, LERW, to SLE(2) when the curves are parametrized by capacity. We construct a coupling of the chordal version of LERW and chordal SLE(2) based on the Green's function for LERW as martingale observable and using an elementary discrete-time Loewner "difference" equation. This coupling is different than the ones previously considered in this context. Our recent work (arXiv:1603.05203) on the convergence of LERW parametrized by length to SLE(2) parameterized by Minkowski content uses specific features of the coupling constructed here.

Related

• Convergence of radial loop-erased random walk in the natural parametrization
• Convergence of loop-erased random walk in the natural parametrization
• Loop-Erased Random Surfaces
• Stochastic Loewner evolution in doubly connected domains
• Boundary behaviour of RW's on planar graphs and convergence of LERW to chordal SLE₂