Skip to main content Accessibility help
×
Home

Line-of-Sight Percolation

  • BÉLA BOLLOBÁS (a1), SVANTE JANSON (a2) and OLIVER RIORDAN (a3)

Abstract

Given ω ≥ 1, let be the graph with vertex set in which two vertices are joined if they agree in one coordinate and differ by at most ω in the other. (Thus is precisely .) Let pc(ω) be the critical probability for site percolation on . Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that limω→∞ωpc(ω)=log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.

Copyright

References

Hide All
[1]Aizenman, M., Kesten, H. and Newman, C. M. (1987) Uniqueness of the infinite cluster and related results in percolation. In Percolation Theory and Ergodic Theory of Infinite Particle Systems, Springer, pp. 1320.
[2]Athreya, K. B. and Ney, P. E. (1972) Branching Processes, Springer, Berlin.
[3]Balister, P., Bollobás, B. and Walters, M. (2004) Continuum percolation with steps in an annulus. Ann. Appl. Probab. 14 18691879.
[4]Balister, P., Bollobás, B. and Walters, M. (2005) Continuum percolation with steps in the square or the disc. Random Struct. Alg. 26 392403.
[5]Bollobás, B., Janson, S. and Riordan, O. (2007) The phase transition in inhomogeneous random graphs. Random Struct. Alg. 31 3122.
[6]Bollobás, B., Janson, S. and Riordan, O. (2007) Spread-out percolation in . Random Struct. Alg. 31 239246.
[7]Bollobás, B. and Riordan, O. (2006) Percolation, Cambridge University Press.
[8]Franceschetti, M., Booth, L., Cook, M., Meester, R. and Bruck, J. (2005) Continuum percolation with unreliable and spread-out connections. J. Statist. Phys. 118 721734.
[9]Frieze, A., Kleinberg, J., Ravi, R. and Debany, W. (2007) Line-of-sight networks. In Proc. 18th ACM–SIAM Symposium on Discrete Algorithms, pp. 968–977.
[10]Frieze, A., Kleinberg, J., Ravi, R. and Debany, W. Line-of-sight networks. Combin. Probab. Comput., to appear.
[11]Gilbert, E. N. (1961) Random plane networks. J. Soc. Indust. Appl. Math. 9 533543.
[12]Glaz, J., Naus, J. and Wallenstein, S. (2001) Scan Statistics, Springer.
[13]Hall, P. (1985) On continuum percolation. Ann. Probab. 13 12501266.
[14]Penrose, M. D. (1993) On the spread-out limit for bond and continuum percolation. Ann. Appl. Probab. 3 253276.
[15]Stevens, W. L. (1939) Solution to a geometrical problem in probability. Ann. Eugenics 9 315320.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed