The Annals of Probability

Planar lattices do not recover from forest fires

Demeter Kiss, Ioan Manolescu, and Vladas Sidoravicius

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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.

Article information

Source
Ann. Probab., Volume 43, Number 6 (2015), 3216-3238.

Dates
Received: January 2014
Revised: July 2014
First available in Project Euclid: 11 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1449843629

Digital Object Identifier
doi:10.1214/14-AOP958

Mathematical Reviews number (MathSciNet)
MR3433580

Zentralblatt MATH identifier
1337.60244

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Keywords
Self-destructive percolation planar percolation critical percolation near-critical percolation forest fires

Citation

Kiss, Demeter; Manolescu, Ioan; Sidoravicius, Vladas. Planar lattices do not recover from forest fires. Ann. Probab. 43 (2015), no. 6, 3216--3238. doi:10.1214/14-AOP958. https://projecteuclid.org/euclid.aop/1449843629


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References

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