Natural parametrization of percolation interface and pivotal points

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.

arXiv comment27 pages, 3 figures. Minor revision according to the referee report. Accepted version. To appear at AIHP

Cited by in corpus (1)

• Liouville dynamical percolation