Planar lattices do not recover from forest fires

Abstract

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 $p>p_c$, $θ(p,δ)=0$. 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.