Natural parametrization of percolation interface and pivotal points

2018/04/19 by Nina Holden, Xinyi Li, Holden, Nina +3 · 1 citation
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1804.07286

openalex publication_date 2018/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We prove that the interface of critical site percolation on the triangular lattice converges to SLE₆ in its natural parametrization, where the discrete interface is parametrized such that each edge is crossed in one unit of time, while the limiting curve is parametrized by the 7/4-dimensional Minkowski content. We also prove that the scaling limit of counting measure on the pivotal points, which was proved to exist by Garban, Pete, and Schramm (2013), is the 3/4-dimensional Minkowski content up to a deterministic multiplicative constant.

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