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

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.

MSC:

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

Zbl 1054.60105
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References:

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