Open Access
November 2015 Planar lattices do not recover from forest fires
Demeter Kiss, Ioan Manolescu, Vladas Sidoravicius
Ann. Probab. 43(6): 3216-3238 (November 2015). DOI: 10.1214/14-AOP958


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.


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Demeter Kiss. Ioan Manolescu. Vladas Sidoravicius. "Planar lattices do not recover from forest fires." Ann. Probab. 43 (6) 3216 - 3238, November 2015.


Received: 1 January 2014; Revised: 1 July 2014; Published: November 2015
First available in Project Euclid: 11 December 2015

zbMATH: 1337.60244
MathSciNet: MR3433580
Digital Object Identifier: 10.1214/14-AOP958

Primary: 60K35 , 82B43

Keywords: Critical percolation , forest fires , Near-critical percolation , planar percolation , Self-destructive percolation

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 6 • November 2015
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