Electronic Journal of Probability

Clustering and percolation of point processes

Bartlomiej Blaszczyszyn and Dhandapani Yogeshwaran

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<p>We are interested in phase transitions in certain percolation models on point processes and their dependence on clustering properties of the point processes. We show that point processes with smaller void probabilities and factorial moment measures than the stationary Poisson point process exhibit non-trivial phase transition in the percolation of some coverage models based on level-sets of additive functionals of the point process. Examples of such point processes are determinantal point processes, some perturbed lattices and more generally, negatively associated point processes. Examples of such coverage models are k-coverage in the Boolean model (coverage by at least k grains), and SINR-coverage (coverage if the signal to-interference-and-noise ratio is large). In particular, we answer in affirmative the hypothesis of existence of phase transition in the percolation of k-faces in the Cech simplicial complex (called also clique percolation) on point processes which cluster less than the Poisson process. We also construct a Cox point process, which is "more clustered” than the Poisson point process and whose Boolean model percolates for arbitrarily small radius. This shows that clustering (at least, as detected by our specific tools) does not always “worsen” percolation, as well as that upper-bounding this cluster-ing by a Poisson process is a consequential assumption for the phase transition to hold.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 72, 20 pp.

Accepted: 31 July 2013
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes 82B43: Percolation [See also 60K35] 60E15
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G60: Random fields

point process Boolean model percolation phase transition shot-noise fields level-sets directionally convex ordering perturbed lattices determinantal sub-Poisson point processes

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Blaszczyszyn, Bartlomiej; Yogeshwaran, Dhandapani. Clustering and percolation of point processes. Electron. J. Probab. 18 (2013), paper no. 72, 20 pp. doi:10.1214/EJP.v18-2468. https://projecteuclid.org/euclid.ejp/1465064297

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