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Percolation and limit theory for the Poisson lilypond model. (English) Zbl 1267.60107

Suppose \(\varphi\) is a locally finite set of points of cardinality at least two in the space \({\mathbb R}^d\). The lilypond model based on \(\varphi\) is the system of balls (or grains) \(\{B_{\rho(x)}(x):\, x\in \varphi\}\) (here, \(B_r(x) =\{y\in {\mathbb R}^d: |y-x|\leq r\}\) and \(|\cdot|\) is the Euclidean norm) with the properties (the hard-core property) \(\rho(x) + \rho(y) \leq |x - y|\) for all different \(x, y\in \varphi\); and (the smaller grain-neighbor property) for each \(x\in \varphi\), there is at least one \(y\in \varphi\setminus\{x\}\) such that \(\rho(x)+ \rho(y) = |y - x|\) (in which case the points \(x\) and \(y\) are called grain-neighbors) and \(\rho(y) \leq \rho(x)\) (\(y\) is called a smaller grain-neighbor of \(x\)). Note that \(\rho(x)= \rho(x, \varphi)\) is a function which depends on both \(\varphi\) and \(x\in \varphi\).
M. Heveling and G. Last [Random Struct. Algorithms 29, No. 3, 338–350 (2006; Zbl 1220.60008)] established the existence and uniqueness of the model for all such \(\varphi\). They showed that \(\rho\) is a measurable function of both arguments. The authors of this paper consider the lilypond model on random \(\varphi\).
The lilypond model was introduced in [O. Häggström and R. Meester, Random Struct. Algorithms 9, No. 3, 295–315 (1996; Zbl 0866.60088)] for the Poisson lilypond model (that is, for a stationary Poisson process \(\Phi\) in \({\mathbb R}^d\) of intensity one). Häggström and Meester proved that the union set \(Z:=Z(\Phi):=\bigcup_{x\in\Phi}B_{\rho(x,\Phi)}(x)\) does not percolate, that is, does not have an unbounded connected component.
Interestingly, there does exist a stationary percolating hard-core system of (non-lilypond) grains on \(\Phi\), at least in high dimensions, see [C. Cotar et al., Random Struct. Algorithms 34, No. 2, 285–299 (2009; Zbl 1159.60351)]. The contributions of the present paper fall into three categories, namely tail bounds (Section 4), central limit theorems (Section 5), and non-percolating under positive enhancement (Section 6). These may be viewed as extending the percolation theory of the lilypond model beyond the basic fact that \(Z\) does not percolate. Sections 2 and 3 develop notions of stabilization.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
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References:

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