## The Annals of Probability

### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.aop/1449843629

Digital Object Identifier
doi:10.1214/14-AOP958

Mathematical Reviews number (MathSciNet)
MR3433580

Zentralblatt MATH identifier
1337.60244

#### 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

#### References

• [1] Ahlberg, D., Duminil-Copin, H., Kozma, G. and Sidoravicius, V. (2013). Seven-dimensional forest fires. Preprint. Available at arXiv:1302.6872.
• [2] Ahlberg, D., Sidoravicius, V. and Tykesson, J. (2013) Bernoulli and self-destructive percolation on non-amenable graphs. Preprint. Available at arXiv:1302.6870.
• [3] Borgs, C., Chayes, J. T., Kesten, H. and Spencer, J. (1999). Uniform boundedness of critical crossing probabilities implies hyperscaling. Random Structures Algorithms 15 368–413.
• [4] Drmota, M. (2009). Random Trees: An Interplay Between Combinatorics and Probability. Springer, New York.
• [5] Drossel, B. and Schwabl, F. (1992). Self-organized critical forest-fire model. Phys. Rev. Lett. 69 1629–1632.
• [6] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
• [7] Kesten, H. (1982). Percolation Theory for Mathematicians. Progress in Probability and Statistics 2. Birkhäuser, Boston, MA.
• [8] Kesten, H. (1986). The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields 73 369–394.
• [9] Kesten, H. (1987). Scaling relations for $2$D-percolation. Comm. Math. Phys. 109 109–156.
• [10] Kiss, D. (2014). Large deviation bounds for the volume of the largest cluster in 2D critical percolation. Electron. Commun. Probab. 19 1–11.
• [11] Kruskal, J. B. Jr. (1956). On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc. 7 48–50.
• [12] Nolin, P. (2008). Near-critical percolation in two dimensions. Electron. J. Probab. 13 1562–1623.
• [13] Ráth, B. and Tóth, B. (2009). Erdős–Rényi random graphs $+$ forest fires $=$ self-organized criticality. Electron. J. Probab. 14 1290–1327.
• [14] Reimer, D. (2000). Proof of the van den Berg–Kesten conjecture. Combin. Probab. Comput. 9 27–32.
• [15] Russo, L. (1978). A note on percolation. Z. Wahrsch. Verw. Gebiete 43 39–48.
• [16] Seymour, P. D. and Welsh, D. J. A. (1978). Percolation probabilities on the square lattice. Ann. Discrete Math. 3 227–245.
• [17] van den Berg, J. and Brouwer, R. (2004). Self-destructive percolation. Random Structures Algorithms 24 480–501.
• [18] van den Berg, J. and Brouwer, R. (2006). Self-organized forest-fires near the critical time. Comm. Math. Phys. 267 265–277.
• [19] van den Berg, J., Brouwer, R. and Vágvölgyi, B. (2008). Box-crossings and continuity results for self-destructive percolation in the plane. In In and Out of Equilibrium 2. Progress in Probability 60 117–135. Birkhäuser, Basel.
• [20] van den Berg, J. and de Lima, B. N. B. (2009). Linear lower bounds for $\delta_{\mathrm{c}}(p)$ for a class of 2D self-destructive percolation models. Random Structures Algorithms 34 520–526.