## The Annals of Probability

### 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 $\mathbb{Z}^{2}$, 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.

#### Article information

Source
Ann. Probab., Volume 43, Number 6 (2015), 2949-3005.

Dates
Revised: June 2014
First available in Project Euclid: 11 December 2015

https://projecteuclid.org/euclid.aop/1449843624

Digital Object Identifier
doi:10.1214/14-AOP950

Mathematical Reviews number (MathSciNet)
MR3433575

Zentralblatt MATH identifier
1341.60128

#### Citation

Hammond, Alan; Pete, Gábor; Schramm, Oded. Local time on the exceptional set of dynamical percolation and the incipient infinite cluster. Ann. Probab. 43 (2015), no. 6, 2949--3005. doi:10.1214/14-AOP950. https://projecteuclid.org/euclid.aop/1449843624

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