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arXiv · 1804.07286

Natural parametrization of percolation interface and pivotal points

Nina Holden·Xinyi Li·Xin Sun
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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.

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Nina Holden, Xinyi Li, Xin Sun. 2018-04-19. Natural parametrization of percolation interface and pivotal points. https://arxiv.org/abs/1804.07286

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