Natural parametrization of percolation interface and pivotal points

TL;DR

This paper establishes that the critical percolation interface on the triangular lattice converges to SLE$_6$ in its natural parametrization, and characterizes the scaling limit of pivotal points as a Minkowski content measure.

Contribution

It proves the convergence of the percolation interface to SLE$_6$ in its natural parametrization and identifies the scaling limit of pivotal points as a Minkowski content measure.

Findings

Percolation interface converges to SLE$_6$ in natural parametrization.

Scaling limit of pivotal points is a Minkowski content measure.

Provides a rigorous link between discrete percolation and SLE$_6$.

Abstract

We prove that the interface of critical site percolation on the triangular lattice converges to SLE$_6$ 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.