Abstract
We consider the supercritical finite-range random connection model where the points x,y of a homogeneous planar Poisson process are connected with probability f(|y−x|) for a given f. Performing percolation on the resulting graph, we show that the critical probabilities for site and bond percolation satisfy the strict inequality \(p_{c}^{\mathrm{site}} > p_{c}^{\mathrm{bond}}\). We also show that reducing the connection function f strictly increases the critical Poisson intensity.
Finally, we deduce that performing a spreading transformation on f (thereby allowing connections over greater distances but with lower probabilities, leaving average degrees unchanged) strictly reduces the critical Poisson intensity. This is of practical relevance, indicating that in many real networks it is in principle possible to exploit the presence of spread-out, long range connections, to achieve connectivity at a strictly lower density value.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aizenman, M., Grimmett, G.: Strict monotonicity for critical points in percolation and ferromagnetic models. J. Stat. Phys. 63, 817–835 (1991)
Balister, P., Bollobás, B., Walters, M.: Continuum percolation with steps in an annulus. Ann. Appl. Probab. 14, 1869–1879 (2004)
Bezuidenhout, C., Grimmett, G., Kesten, H.: Strict inequality for critical values of Potts models and random-cluster processes. Commun. Math. Phys. 158, 1–16 (1993)
Bollobás, B., Janson, S., Riordan, O.: Spread-out percolation in ℝd. Random Struct. Algorithms 31, 239–246 (2007)
Franceschetti, M., Meester, R.: Random Networks for Communication. Cambridge University Press, Cambridge (2007)
Franceschetti, M., Booth, L., Bruck, J., Meester, R.: Continuum percolation with unreliable and spread-out connections. J. Stat. Phys. 118, 721–734 (2005)
Grimmett, G.: Potts models and Random-Cluster models with many-body interactions. J. Stat. Phys. 75, 67–121 (1994)
Grimmett, G., Stacey, A.: Critical probabilities for site and bond percolation models. Ann. Probab. 26, 1788–1812 (1998)
Jonasson, J.: Optimization of shape in continuum percolation. Ann. Probab. 29, 624–635 (2001)
Meester, R., Roy, R.: Continuum Percolation. Cambridge University Press, Cambridge (1996)
Menshikov, M.V.: Quantitative estimates and rigorous inequalities for critical points of a graph and its subgraphs. Theory Probab. Appl. 32, 544–547 (1987)
Penrose, M.D.: On a continuum percolation model. Adv. Appl. Probab. 23, 536–556 (1991)
Penrose, M.D.: On the spread-out limit for bond and continuum percolation. Ann. Appl. Probab. 3, 253–276 (1993)
Penrose, M.D.: Random Geometric Graphs. Oxford University Press, London (2003)
Penrose, M.D., Meester, R., Sarkar, A.: The random connection model in high dimensions. Stat. Probab. Lett. 35, 145–153 (1997)
Roy, R., Sarkar, A., White, D.: Backbends in directed percolation. J. Stat. Phys. 91, 889–908 (1998)
Roy, R., Tanemura, H.: Critical intensities of Boolean models with different underlying convex shapes. Adv. Appl. Probab. 34, 48–57 (2002)
Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 229–237 (1981)
Sarkar, A.: Co-existence of the occupied and vacant phase in Boolean models in three or more dimensions. Adv. Appl. Probab. 29, 878–889 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of M. Franceschetti was partially supported by NSF grant CNS0916778.
The research of M.D. Penrose was partially supported by the Isaac Newton Institute, Cambridge, UK.
The research of T. Rosoman was supported by an EPSRC studentship.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Franceschetti, M., Penrose, M.D. & Rosoman, T. Strict Inequalities of Critical Values in Continuum Percolation. J Stat Phys 142, 460–486 (2011). https://doi.org/10.1007/s10955-011-0122-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-011-0122-1