## The Annals of Applied Probability

### On the capacity functional of the infinite cluster of a Boolean model

#### Abstract

Consider a Boolean model in $\mathbb{R}^{d}$ with balls of random, bounded radii with distribution $F_{0}$, centered at the points of a Poisson process of intensity $t>0$. The capacity functional of the infinite cluster $Z_{\infty}$ is given by $\theta_{L}(t)=\mathbb{P}\{Z_{\infty}\cap L\ne\varnothing \}$, defined for each compact $L\subset\mathbb{R}^{d}$.

We prove for any fixed $L$ and $F_{0}$ that $\theta_{L}(t)$ is infinitely differentiable in $t$, except at the critical value $t_{c}$; we give a Margulis–Russo-type formula for the derivatives. More generally, allowing the distribution $F_{0}$ to vary and viewing $\theta_{L}$ as a function of the measure $F:=tF_{0}$, we show that it is infinitely differentiable in all directions with respect to the measure $F$ in the supercritical region of the cone of positive measures on a bounded interval.

We also prove that $\theta_{L}(\cdot)$ grows at least linearly at the critical value. This implies that the critical exponent known as $\beta$ is at most 1 (if it exists) for this model. Along the way, we extend a result of Tanemura [J. Appl. Probab. 30 (1993) 382–396], on regularity of the supercritical Boolean model in $d\geq3$ with fixed-radius balls, to the case with bounded random radii.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 3 (2017), 1678-1701.

Dates
Revised: July 2016
First available in Project Euclid: 19 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1500451238

Digital Object Identifier
doi:10.1214/16-AAP1241

Mathematical Reviews number (MathSciNet)
MR3678482

Zentralblatt MATH identifier
1373.60160

#### Citation

Last, Günter; Penrose, Mathew D.; Zuyev, Sergei. On the capacity functional of the infinite cluster of a Boolean model. Ann. Appl. Probab. 27 (2017), no. 3, 1678--1701. doi:10.1214/16-AAP1241. https://projecteuclid.org/euclid.aoap/1500451238

#### References

• Bollobás, B. (1979). Graph Theory: An Introductory Course. Springer, New York.
• Chayes, J. T. and Chayes, L. (1986). Inequality for the infinite-cluster density in Bernoulli percolation. Phys. Rev. Lett. 56 1619–1622.
• Duminil-Copin, H. and Tassion, V. (2015). A new proof of sharpness of the phase transition for Bernoulli percolation on ${\mathbb{Z}}^{d}$. Enseign. Math. 62 199–206.
• Gouéré, J. B. (2008). Subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Probab. 36 1209–1220.
• Gouéré, J. B. and Marchand, R. (2011). Continuum percolation in high dimensions. Technical report, available at arXiv:1108.6133.
• Grimmett, G. (1999). Percolation, 2nd ed. Springer, Berlin.
• Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A 430 439–457.
• Gupta, J. and Rao, B. (1999). Van den Berg–Kesten inequality for the Poisson Boolean model for continuum percolation. Sankhya, Ser. A 61 337–346.
• Hall, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York.
• Jiang, J., Zhang, S. and Guo, T. (2011). Russo’s formula, uniqueness of the infinite cluster, and continuous differentiability of free energy for continuum percolation. J. Appl. Probab. 48 597–610.
• Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
• Last, G. (2014). Perturbation analysis of Poisson processes. Bernoulli 20 486–513.
• Last, G. and Penrose, M. (2017). Lectures on the Poisson Process. Cambridge Univ. Press, Cambridge. To appear.
• Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Univ. Press, Cambridge.
• Meester, R., Roy, R. and Sarkar, A. (1994). Nonuniversality and continuity of the critical covered volume fraction in continuum percolation. J. Stat. Phys. 75 123–134.
• Molchanov, I. (2005). Theory of Random Sets. Springer, London.
• Molchanov, I. and Zuyev, S. (2000). Variational analysis of functionals of Poisson processes. Math. Oper. Res. 25 485–508.
• Penrose, M. (2003). Random Geometric Graphs. Oxford Univ. Press, London.
• Penrose, M. D. (2007). Gaussian limits for random geometric measures. Electron. J. Probab. 12 989–1035.
• Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
• Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications. Wiley, Chichester.
• Tanemura, H. (1993). Behavior of the supercritical phase of a continuum percolation model on ${\mathbf{R}}^{d}$. J. Appl. Probab. 30 382–396.
• Zuev, S. A. (1992). Russo’s formula for the Poisson point processes and its applications. Diskret. Mat. 4 149–160.
• Zuev, S. A. and Sidorenko, A. F. (1985). Continuous models of percolation theory II. Teotretich. i Matematich. Fizika 62 253–262.