## Annals of Probability

### Nonoptimality of constant radii in high dimensional continuum percolation

#### Abstract

Consider a Boolean model $\Sigma$ in $\mathbb{R}^{d}$. The centers are given by a homogeneous Poisson point process with intensity $\lambda$ and the radii of distinct balls are i.i.d. with common distribution $\nu$. The critical covered volume is the proportion of space covered by $\Sigma$ when the intensity $\lambda$ is critical for percolation. Previous numerical simulations and heuristic arguments suggest that the critical covered volume may be minimal when $\nu$ is a Dirac measure. In this paper, we prove that it is not the case in sufficiently high dimension.

#### Article information

Source
Ann. Probab., Volume 44, Number 1 (2016), 307-323.

Dates
Revised: September 2014
First available in Project Euclid: 2 February 2016

https://projecteuclid.org/euclid.aop/1454423042

Digital Object Identifier
doi:10.1214/14-AOP974

Mathematical Reviews number (MathSciNet)
MR3456339

Zentralblatt MATH identifier
1342.60167

#### Citation

Gouéré, Jean-Baptiste; Marchand, Régine. Nonoptimality of constant radii in high dimensional continuum percolation. Ann. Probab. 44 (2016), no. 1, 307--323. doi:10.1214/14-AOP974. https://projecteuclid.org/euclid.aop/1454423042

#### References

• [1] Balram, A. and Dhar, D. (2010). Scaling relation for determining the critical threshold for continuum percolation of overlapping discs of two sizes. Pramana 74 109–114.
• [2] Consiglio, R., Baker, D. R., Paul, G. and Stanley, H. E. (2003). Continuum percolation thresholds for mixtures of spheres of different sizes. Physica A: Statistical Mechanics and Its Applications 319 49–55.
• [3] Dhar, D. (1997). On the critical density for continuum percolation of spheres of variable radii. Physica A: Statistical and Theoretical Physics 242 341–346.
• [4] Dhar, D. and Phani, M. K. (1984). Continuum percolation with discs having a distribution of radii. J. Phys. A 17 L645–L649.
• [5] Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Probab. 12 999–1040.
• [6] Gouéré, J.-B. (2008). Subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Probab. 36 1209–1220.
• [7] Gouéré, J.-B. (2014). Percolation in a multiscale Boolean model. ALEA Lat. Am. J. Probab. Math. Stat. 11 281–297.
• [8] Gouéré, J.-B. and Marchand, R. (2013). Continuum percolation in high dimension. Available at arXiv:1108.6133.
• [9] Kertész, J. and Vicsek, T. (1982). Monte Carlo renormalization group study of the percolation problem of discs with a distribution of radii. Zeitschrift für Physik B Condensed Matter 45 345–350.
• [10] Kesten, H. (1990). Asymptotics in high dimensions for percolation. In Disorder in Physical Systems 219–240. Oxford Univ. Press, New York.
• [11] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Tracts in Mathematics 119. Cambridge Univ. Press, Cambridge.
• [12] 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.
• [13] Menshikov, M. V., Popov, S. Yu. and Vachkovskaia, M. (2001). On the connectivity properties of the complementary set in fractal percolation models. Probab. Theory Related Fields 119 176–186.
• [14] Penrose, M. D. (1996). Continuum percolation and Euclidean minimal spanning trees in high dimensions. Ann. Appl. Probab. 6 528–544.
• [15] Quintanilla, J. A. and Ziff, R. M. (2007). Asymmetry in the percolation thresholds of fully penetrable disks with two different radii. Phys. Rev. E 76 051115.