Local time on the exceptional set of dynamical percolation and the incipient infinite cluster

• Hammond, Alan ^[1] ; Pete, Gábor ^[3] ; Schramm, Oded ^[2]
  1. [1] University of Oxford

     University of Oxford

     Oxford District, Reino Unido

     
  2. [2] The Microsoft Research - University of Trento Centre for Computational and Systems Biology

     The Microsoft Research - University of Trento Centre for Computational and Systems Biology

     Trento, Italia

     
  3. [3] Technical University of Budapest

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 6, 2015, págs. 2949-3005
• Idioma: inglés
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• Resumen

  • In dynamical critical site percolation on the triangular lattice or bond percolation on Z2, 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.