Skip to main content Accessibility help
×
Home
Hostname: page-component-dc8c957cd-fcmtc Total loading time: 0.38 Render date: 2022-01-28T03:31:12.312Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

On the critical threshold for continuum AB percolation

Published online by Cambridge University Press:  16 January 2019

David Dereudre*
Affiliation:
Université de Lille
Mathew Penrose*
Affiliation:
University of Bath
*
* Postal address: Laboratoire Paul Painlevé, Université de Lille, 59655 Villeneuve d’Ascq, France. Email address: david.dereudre@univ-lille.fr
** Postal address: Department of Mathematical Sciences, University of Bath, BathBA2 7AY, UK.

Abstract

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ,μ. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c∕(λ−λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μcc) =∞.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Esary, J. D. and Proschan, F. (1963). Coherent structures of non-identical components. Technometrics 5, 191209.CrossRefGoogle Scholar
[2]Franceschetti, M., Pensure, M. D. and Rosoman, T. (2010). Strict inequalities of critical probabilities on Gilbert’s continuum percolation graph. Preprint. Available at https://arxiv.org/abs/1004.1596.Google Scholar
[3]Franceschetti, M., Pensure, M. D. and Rosoman, T. (2011). Strict inequalities of critical values in continuum percolation. J. Statist. Phys. 142, 460486.CrossRefGoogle Scholar
[4]Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
[5]Iyer, S. K. and Yogeshwaran, D. (2012). Percolation and connectivity in AB random geometric graphs. Adv. Appl. Prob. 44, 2141.CrossRefGoogle Scholar
[6]Last, G. and Penrose, M. (2018). Lectures on the Poisson Process. Cambridge University Press.Google Scholar
[7]Lorenz, C. D. and Ziff, R. M. (2001). Precise determination of the critical percolation threshold for the three dimensional “Swiss cheese” model using a growth algorithm. J. Chem. Phys. 114, 36593661.CrossRefGoogle Scholar
[8]Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.CrossRefGoogle Scholar
[9]Penrose, M. D. (2014). Continuum AB percolation and AB random geometric graphs. J. Appl. Prob. 51A, 333344.CrossRefGoogle Scholar
[10]Pinti, P. C. and Win, Z. (2012). Percolation and connectivity in the intrinsically secure communications graph. IEEE Trans. Inform. Theory 58, 17161730.CrossRefGoogle Scholar
[11]Quintanilla, J. A. and Ziff, R. M. (2007). Asymmetry of percolation thresholds of fully penetrable disks with two different radii. Phys. Rev. E 76, 051115.CrossRefGoogle ScholarPubMed
[12]Sarkar, A. and Haenggi, M. (2013). Percolation in the Secrecy Graph. Discrete Appl. Math. 161, 21202132.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On the critical threshold for continuum AB percolation
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

On the critical threshold for continuum AB percolation
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

On the critical threshold for continuum AB percolation
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *