The rate of convergence of harmonic explorer to SLE4

2020/03/04 by Shiyi Lan, Jin Ma, Lan, Shi-Yi +3
Biochemistry, Genetics and Molecular Biology · Economics, Econometrics and Finance · Mathematics · #60D05 #60J65 #82B41 #Complex Variables (math.CV) #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2003.01949

openalex publication_date 2020/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using the estimate of the difference between the discrete harmonic function and its corresponding continuous version we derive a rate of convergence of the Loewner driving function for the harmonic explorer to the Brownian motion with speed 4 on the real line. Based on this convergence rate, the derivative estimate for chordal SLE₄, and the estimate of tip structure modulus for harmonic explorer paths, we obtain an explicit power-law rate of convergence of the harmonic explorer paths to the trace of chordal SLE₄ in the supremum distance.

Citations

• Contour lines of the two-dimensional discrete Gaussian free field

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