Planar lattices do not recover from forest fires

Kiss, Demeter ^[1] ; Manolescu, Ioan ^[2] ; Sidoravicius, Vladas ^[3]
1. [1] University of Cambridge
  
  University of Cambridge
  
  Cambridge District, Reino Unido
2. [2] Tohoku University
  
  Tohoku University
  
  Aoba-ku, Japón
3. [3] Université de Genève
  
  Université de Genève
  
  Genève, Suiza
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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. 3216-3238
Idioma: inglés
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Resumen
- Self-destructive percolation with parameters p,δ is obtained by taking a site percolation configuration with parameter p, closing all sites belonging to infinite clusters, then opening every closed site with probability δ, independently of the rest. Call θ(p,δ) the probability that the origin is in an infinite cluster in the configuration thus obtained.
  
  For two-dimensional lattices, we show the existence of δ>0 such that, for any p>pc, θ(p,δ)=0. This proves the conjecture of van den Berg and Brouwer [Random Structures Algorithms 24 (2004) 480–501], who introduced the model. Our results combined with those of van den Berg and Brouwer [Random Structures Algorithms 24 (2004) 480–501] imply the nonexistence of the infinite parameter forest-fire model. The methods herein apply to site and bond percolation on any two-dimensional planar lattice with sufficient symmetry.