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Planar lattices do not recover from forest fires

Thursday 23rd April 2015 - 14:00 to 15:00
INI Seminar Room 1
Co-authors: Demeter Kiss (University of Cambridge) and Vladas Sidoravicius (IMPA)

Self-destructive percolation with parameters p, delta is obtained by taking a site percolation configuration with parameter p, closing all sites belonging to the infinite cluster, then opening every site with probability delta, independently of the rest. Call theta(p,delta) the probability that the origin is in an infinite cluster in the configuration thus obtained. For two dimensional lattices, we show the existence of delta > 0 such that, for any p > p_c , theta(p,delta) = 0. This proves a conjecture of van den Berg and Brouwer, who introduced the model. Our results also imply the non-existence of the infinite parameter forest-fire model on planar lattices.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons