Local time on the exceptional set of dynamical percolation, and the Incipient Infinite Cluster
Abstract
In dynamical critical site percolation on the triangular lattice or bond percolation on \Z^2, we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster. We show that at a typical time with respect to this measure, the percolation configuration has the law of Kesten's Incipient Infinite Cluster. In the most technical result of this paper, we show that, on the other hand, at the first exceptional time, the law of the configuration is different. We also study the collapse of the infinite cluster near typical exceptional times, and establish a relation between static and dynamic exponents, analogous to Kesten's near-critical relation.
Keywords
Cite
@article{arxiv.1208.3826,
title = {Local time on the exceptional set of dynamical percolation, and the Incipient Infinite Cluster},
author = {Alan Hammond and Gábor Pete and Oded Schramm},
journal= {arXiv preprint arXiv:1208.3826},
year = {2013}
}
Comments
56 pages, 7 figures. In this version, we prove convergence only for one of our local time constructions. Consequently, some arguments are slightly changed in Sections 2 and 3