Planar lattices do not recover from forest fires

Abstract

Self-destructive percolation with parameters $p,\delta$ 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 $\delta$, 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 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.