Nonoptimality of constant radii in high dimensional continuum percolation.

*(English)*Zbl 1342.60167The paper under review proves that the critical covered volume cannot be minimized for sufficiently high dimensions when the common distribution is a Dirac measure. This implies that the Kertesz-Vicsek conjecture is false in high dimensions.

The Boolean model \(\Sigma (\lambda \nu)\) in \(\mathbb R^d\) driven by \(\lambda \nu\) is the random subset \(\cup_{(c, r)\in \xi} B(c, r)\), where \(B(c,r)\) is the open Euclidean ball centered at \(c\in \mathbb R^d\) with radius \(r\in (0,\infty)\), for a Poisson point process \(\xi\) on \(\mathbb R^d \times (0, \infty)\). The Boolean model \(\Sigma (\lambda \nu)\) percolates if the probability that there is an unbounded connected component of \(\Sigma (\lambda \nu)\) that contains the origin is positive, and the critical intensity is defined by \(\lambda^c_d(\nu) = \inf \{\lambda> 0: \Sigma (\lambda \nu)\, \text{percolates}\}\). The critical covered volume through the normalized critical intensity is \(\tilde{\lambda}_d^c(\nu) = \lambda^c_d(\nu)\int \nu_d(2r)^d \nu(dr)\). J. Kertész and T. Vicsek conjectured in [Z. Phys., B 45, No. 4, 345–350 (1982; Zbl 1342.60170)] that the normalized critical intensity should be independent of \(\nu\) as long as supp(\(\nu\)) is bounded. R. Meester et al. gave a rigorous counterexample for this conjecture in [J. Stat. Phys. 75, No. 1–2, 123–134 (1994; Zbl 0828.60083)]. There is numerical evidence that the conjecture is true for \(d=2, 3\) with a deterministic radius. For the deterministic radius, M. D. Penrose [Ann. Appl. Probab. 6, No. 2, 528–544 (1996; Zbl 0855.60096)] proved \(\lim_{d\to \infty}\tilde{\lambda}_d^c(\delta_1) =1\), where \(\delta_1\) is the Dirac measure.

Consider the two-type Boolean model \(\Sigma = \Sigma_1 \cup \Sigma_{\rho}\) with \(\Sigma_1 = \cup_{x\in \chi_1}B(x, 1)\) and \(\Sigma_{\rho} = \cup_{x\in \chi_{\rho}} B(x, \rho)\), and its measure \(\lambda_1\delta_1 + \lambda_{\rho}\delta_{\rho} = \frac{k^d \mu_d}{\nu_d2^d}\) (for \(\mu_d = \delta_1 + \frac{\delta_{\rho}}{\rho^d}\) and \(k^c_{\rho} = \frac{2\sqrt{\rho}}{1+\rho}<1\)). Proposition 2.1 and Proposition 2.2 show that the percolation does not occur in the two-type Boolean model for the subcritical phase \(k<k^c_{\rho}\) and the supercritical phase \(k>k^c_{\rho}\), respectively, for sufficiently large dimensions \(d\). The proof for the subcritical phase is similar to the proof of Penrose’s theorem for the two-type Galton-Watson process instead of the one-type Galton-Watson process. The main part of the paper is devoted to the proof for the supercritical phase via R. Durrett’s critical parameter (from [Ann. Probab. 12, 999–1040 (1984; Zbl 0567.60095)]) for oriented percolation in dimension 2 and the Poisson law with parameter (where dimension issues come in) with the assistance of Proposition 2.3.

The main theorem (Theorem 1.2) states that \[ \lim_{d\to \infty} \frac{\ln \tilde{\lambda}_d^c(\mu_d)}{d} = \ln k^c_{\rho}. \] Its proof follows from Proposition 2.1 for the subcritical phase with \(\lim_{d\to \infty} \inf \frac{\ln \tilde{\lambda}_d^c(\mu_d)}{d} \geq \ln k^c_{\rho}\), and from Proposition 2.2 for the supercritical phase with \(\lim_{d\to \infty} \sup \frac{\ln \tilde{\lambda}_d^c(\mu_d)}{d} \leq \ln k^c_{\rho}\) for sufficiently large \(d\). Corollary 1.3 shows that the Kertesz-Vicsek conjecture is false for sufficiently large \(d\) by Penrose’s theorem and Theorem 1.2 (to have the existence of a probability measure \(\nu\) on \((0, \infty)\) such that \(c_d^c(\nu)<c_d^c(\delta_1)\), depending on the measures).

It is unclear if the conjecture is actually true for low dimensions \(d=2, 3\), or if there is a critical dimension to split the conjecture to be true on one part of the dimension region and false on the other part.

The Boolean model \(\Sigma (\lambda \nu)\) in \(\mathbb R^d\) driven by \(\lambda \nu\) is the random subset \(\cup_{(c, r)\in \xi} B(c, r)\), where \(B(c,r)\) is the open Euclidean ball centered at \(c\in \mathbb R^d\) with radius \(r\in (0,\infty)\), for a Poisson point process \(\xi\) on \(\mathbb R^d \times (0, \infty)\). The Boolean model \(\Sigma (\lambda \nu)\) percolates if the probability that there is an unbounded connected component of \(\Sigma (\lambda \nu)\) that contains the origin is positive, and the critical intensity is defined by \(\lambda^c_d(\nu) = \inf \{\lambda> 0: \Sigma (\lambda \nu)\, \text{percolates}\}\). The critical covered volume through the normalized critical intensity is \(\tilde{\lambda}_d^c(\nu) = \lambda^c_d(\nu)\int \nu_d(2r)^d \nu(dr)\). J. Kertész and T. Vicsek conjectured in [Z. Phys., B 45, No. 4, 345–350 (1982; Zbl 1342.60170)] that the normalized critical intensity should be independent of \(\nu\) as long as supp(\(\nu\)) is bounded. R. Meester et al. gave a rigorous counterexample for this conjecture in [J. Stat. Phys. 75, No. 1–2, 123–134 (1994; Zbl 0828.60083)]. There is numerical evidence that the conjecture is true for \(d=2, 3\) with a deterministic radius. For the deterministic radius, M. D. Penrose [Ann. Appl. Probab. 6, No. 2, 528–544 (1996; Zbl 0855.60096)] proved \(\lim_{d\to \infty}\tilde{\lambda}_d^c(\delta_1) =1\), where \(\delta_1\) is the Dirac measure.

Consider the two-type Boolean model \(\Sigma = \Sigma_1 \cup \Sigma_{\rho}\) with \(\Sigma_1 = \cup_{x\in \chi_1}B(x, 1)\) and \(\Sigma_{\rho} = \cup_{x\in \chi_{\rho}} B(x, \rho)\), and its measure \(\lambda_1\delta_1 + \lambda_{\rho}\delta_{\rho} = \frac{k^d \mu_d}{\nu_d2^d}\) (for \(\mu_d = \delta_1 + \frac{\delta_{\rho}}{\rho^d}\) and \(k^c_{\rho} = \frac{2\sqrt{\rho}}{1+\rho}<1\)). Proposition 2.1 and Proposition 2.2 show that the percolation does not occur in the two-type Boolean model for the subcritical phase \(k<k^c_{\rho}\) and the supercritical phase \(k>k^c_{\rho}\), respectively, for sufficiently large dimensions \(d\). The proof for the subcritical phase is similar to the proof of Penrose’s theorem for the two-type Galton-Watson process instead of the one-type Galton-Watson process. The main part of the paper is devoted to the proof for the supercritical phase via R. Durrett’s critical parameter (from [Ann. Probab. 12, 999–1040 (1984; Zbl 0567.60095)]) for oriented percolation in dimension 2 and the Poisson law with parameter (where dimension issues come in) with the assistance of Proposition 2.3.

The main theorem (Theorem 1.2) states that \[ \lim_{d\to \infty} \frac{\ln \tilde{\lambda}_d^c(\mu_d)}{d} = \ln k^c_{\rho}. \] Its proof follows from Proposition 2.1 for the subcritical phase with \(\lim_{d\to \infty} \inf \frac{\ln \tilde{\lambda}_d^c(\mu_d)}{d} \geq \ln k^c_{\rho}\), and from Proposition 2.2 for the supercritical phase with \(\lim_{d\to \infty} \sup \frac{\ln \tilde{\lambda}_d^c(\mu_d)}{d} \leq \ln k^c_{\rho}\) for sufficiently large \(d\). Corollary 1.3 shows that the Kertesz-Vicsek conjecture is false for sufficiently large \(d\) by Penrose’s theorem and Theorem 1.2 (to have the existence of a probability measure \(\nu\) on \((0, \infty)\) such that \(c_d^c(\nu)<c_d^c(\delta_1)\), depending on the measures).

It is unclear if the conjecture is actually true for low dimensions \(d=2, 3\), or if there is a critical dimension to split the conjecture to be true on one part of the dimension region and false on the other part.

Reviewer: Weiping Li (Stillwater)

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B43 | Percolation |

##### Keywords:

continuum percolation; two-type Boolean model; critical covered volume; normalized critical intensity##### References:

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