Planar lattices do not recover from forest fires. (English) Zbl 1337.60244

Kiss, Demeter; Manolescu, Ioan; Sidoravicius, Vladas

Ann. Probab. 43, No. 6, 3216-3238 (2015).

The percolation model treated in this paper is formulated as follows: \(\Omega=\{0,1\}^{V^{{\mathcal Z}^2}}\) is a set of configurations \(\omega=(\omega(x))\) on a planar lattice \({\mathcal Z}^2\) with the set of vertices \(V^{{\mathcal Z}^2}=\{x\}\), where \(w(x)=1\) if \(x\) is open (= \(\omega\)-open) and \(w(x)=0\) if \(x\) is closed (= \(\omega\)-closed). \(x,y\in V^{{\mathcal Z}^2}\) are called connected in a configuration \(\omega\) if there is a self-avoiding \(\omega\)-path with end-points \(x\) and \(y\). Only self-avoiding paths are considered in the paper and a cluster is defined as a connected component of the graph \({\mathcal Z}^2\) formed by open sites. On the set \(\Omega\), a product measure \({\mathcal P}_p\) is defined with \({\mathcal P}_p(\omega(x)=1)=p,\;p\in [0,1]\), for all \(x\in V^{{\mathcal Z}^2}\). Furthermore, for each configuration \(\omega\in \Omega\) and all \(x\in V^{{\mathcal Z}^2}\), a modified configuration \(\bar{\omega}\) is defined as \[ \bar{\omega}(x)=\begin{cases} 1, & \text{\;if} \;\omega(x)=1\;\text{and}\;x\nrightarrow \infty\\ 0, & \text{otherwise.} \end{cases} \] The last definition mimics forest fires, when after appearance of an infinite cluster of trees affected by a fire all trees in the infinite cluster are burned. It is assumed that each burned site is opened with the same probability \(\delta>0\), independently of other sites, and that \(\theta(p,\delta)\) is the probability that the origin belongs to an infinite cluster in a thus obtained configuration. Finally, let \(p_c:=\sup\{p\geq 0: {\mathcal P}_p(0\rightarrow \infty)=0\}\) denote the critical probability. The authors’ main result states that there exists a \(\delta>0\) such that for any \(p>p_c\), \(\theta(p,\delta)=0\). The model was introduced by J. van den Berg and R. Brouwer in [Random Struct. Algorithms 24, No. 4, 480–501 (2004; Zbl 1054.60105)], where they conjectured the aforementioned result.

Reviewer: Boris Granovsky (Haifa)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B43	Percolation

Keywords:

self-destructive percolation; planar percolation; critical percolation; near-critical percolation; forest fires

Citations:

Zbl 1054.60105

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