Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $\mathbb{R}^{d}$

Sebastian Ziesche

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider the Boolean model $Z$ on $\mathbb{R}^{d}$ with random compact grains of bounded diameter, i.e. $Z:=\bigcup_{i\in\mathbb{N}}(Z_{i}+X_{i})$ where $\{X_{1},X_{2},\dots\}$ is a Poisson point process of intensity $t$ and $(Z_{1},Z_{2},\dots)$ is an i.i.d. sequence of compact grains (not necessarily balls) with diameters a.s. bounded by some constant. We will show that exponential decay holds in the sub-critical regime, that means the volume and radius of the cluster of the typical grain in $Z$ have an exponential tail. To achieve this we adapt the arguments of (A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb{Z}^{d}$ (2015) Preprint) and apply a new construction of the cluster of the typical grain together with arguments related to branching processes.

In the second part of the paper, we obtain new lower bounds for the Boolean model with deterministic grains. Some of these bounds are rigorous, while others are obtained via simulation. The simulated bounds come with confidence intervals and are much more precise than the rigorous ones. They improve known results (J. Chem. Phys. 137 (2012) 074106) in dimension six and above.

Résumé

Nous considérons le modèle Booléen $Z$ sur $\mathbb{R}^{d}$ avec des grains compacts aléatoires de diamètres bornés, c’est-à-dire $Z:=\bigcup_{i\in\mathbb{N}}(Z_{i}+X_{i})$ où $\{X_{1},X_{2},\dots\}$ est un processus de Poisson d’intensité $t$ et $(Z_{1},Z_{2},\dots)$ est une suite i.i.d. de grains compacts (non nécessairement des boules) de diamètres p.s. bornés par une constante. Nous montrons une décroissance exponentielle dans le régime sous-critique, ce qui veut dire que le volume et le rayon du cluster d’un grain typique dans $Z$ a une queue exponentielle. Pour cela, nous adaptons des résultats de (A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb{Z}^{d}$ (2015) Preprint) et appliquons une nouvelle construction du cluster d’un grain typique avec des arguments issus des processus de branchement. Dans la seconde partie du papier, nous obtenons de nouvelles bornes inférieures pour le modèle booléen avec grains déterministes. Certaines des ces bornes sont rigoureuses, alors que d’autres sont obtenues par simulation. Les bornes obtenues par simulation sont fournies avec des intervalles de confiance et sont beaucoup plus précises que celles obtenues rigoureusement. Elles améliorent les résultats connus (J. Chem. Phys. 137 (2012) 074106) en dimension 6 et plus.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 2 (2018), 866-878.

Dates
Received: 23 August 2016
Revised: 10 February 2017
Accepted: 6 March 2017
First available in Project Euclid: 25 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1524643232

Digital Object Identifier
doi:10.1214/17-AIHP824

Mathematical Reviews number (MathSciNet)
MR3795069

Zentralblatt MATH identifier
06897971

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G55: Point processes

Keywords
Boolean model Gilbert graph Poisson process Exponential decay Continuum percolation Lower bound Critical intensity

Citation

Ziesche, Sebastian. Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $\mathbb{R}^{d}$. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 866--878. doi:10.1214/17-AIHP824. https://projecteuclid.org/euclid.aihp/1524643232


Export citation

References

  • [1] D. Ahlberg, V. Tassion and A. Teixeira. Sharpness of the phase transition for continuum percolation in $\mathbb{R}^{2}$. Preprint, 2016. Available at arXiv:1605.05926.
  • [2] M. Aizenman and D. J. Barsky. Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 (3) (1987) 489–526.
  • [3] K. B. Athreya and P. E. Ney. Branching Processes, Die Grundlehren der mathematischen Wissenschaften Band 196. Springer-Verlag, New York-Heidelberg, 1972.
  • [4] F. Barthe, O. Guédon, S. Mendelson and A. Naor. A probabilistic approach to the geometry of the $L_{p}$-ball. Ann. Probab. 33 (2) (2005) 480–513.
  • [5] H. Duminil-Copin and V. Tassion. A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb{Z}^{d}$. Enseign. Math. 62 (2016) 199–206.
  • [6] P. Hall. On continuum percolation. Ann. Probab. 13 (4) (1985) 1250–1266.
  • [7] G. Last. Perturbation analysis of Poisson processes. Bernoulli 20 (2) (2014) 486–513.
  • [8] G. Last and M. Penrose. Lectures on the Poisson Process. To be published by Cambridge University Press, 2017. Available at http://www.math.kit.edu/stoch/~last/page/lehrbuch_poissonp/en.
  • [9] S. Li. Concise formulas for the area and volume of a hyperspherical cap. Asian J. Math. Stat. 4 (1) (2011) 66–70.
  • [10] R. Meester and R. Roy. Continuum Percolation. Cambridge Tracts in Mathematics 119. Cambridge University Press, Cambridge, 1996.
  • [11] M. V. Menshikov. Coincidence of critical-points in the percolation problems. Dokl. Akad. Nauk SSSR 288 (6) (1986) 1308–1311.
  • [12] M. D. Penrose. Continuum percolation and Euclidean minimal spanning trees in high dimensions. Ann. Appl. Probab. 6 (2) (1996) 528–544.
  • [13] M. D. Penrose. Random Geometric Graphs. Oxford Studies in Probability 5. Oxford University Press, Oxford, 2003.
  • [14] R. Schneider and W. Weil. Stochastic and Integral Geometry. Probability and its Applications (New York). Springer-Verlag, Berlin, 2008.
  • [15] S. Torquato and Y. Jiao. Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses. J. Chem. Phys. 137 (7) 2012 074106.