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November 2015 Local time on the exceptional set of dynamical percolation and the incipient infinite cluster
Alan Hammond, Gábor Pete, Oded Schramm
Ann. Probab. 43(6): 2949-3005 (November 2015). DOI: 10.1214/14-AOP950

Abstract

In dynamical critical site percolation on the triangular lattice or bond percolation on $\mathbb{Z}^{2}$, 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 believe that the two laws are mutually singular, but do not show this. 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.

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Alan Hammond. Gábor Pete. Oded Schramm. "Local time on the exceptional set of dynamical percolation and the incipient infinite cluster." Ann. Probab. 43 (6) 2949 - 3005, November 2015. https://doi.org/10.1214/14-AOP950

Information

Received: 1 April 2013; Revised: 1 June 2014; Published: November 2015
First available in Project Euclid: 11 December 2015

zbMATH: 1341.60128
MathSciNet: MR3433575
Digital Object Identifier: 10.1214/14-AOP950

Subjects:
Primary: 60K35, 82B27, 82B43
Secondary: 60D05, 60J67

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 6 • November 2015
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