Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T12:19:22.370Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics for First-Passage Times on Delaunay Triangulations

Published online by Cambridge University Press:  03 February 2011

LEANDRO P. R. PIMENTEL
LEANDRO P. R. PIMENTEL*
Affiliation:
Institute of Mathematics, Federal University of Rio de Janeiro, Brazil (e-mail: leandro@im.ufrj.br)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study planar first-passage percolation (FPP) models on random Delaunay triangulations. In [14], Vahidi-Asl and Wierman showed, using sub-additivity theory, that the rescaled first-passage time converges to a finite and non-negative constant μ. We show a sufficient condition to ensure that μ>0 and derive some upper bounds for fluctuations. Our proofs are based on percolation ideas and on the method of martingales with bounded increments.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 3 , May 2011 , pp. 435 - 453
Copyright
Copyright © Cambridge University Press 2011

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]Baik, J., Deift, P. and Johansson, K. (1999) On the distribution of the longest increasing subsequence in a random permutation. J. Amer. Math. Soc. 12 1119–1178.CrossRefGoogle Scholar
[2]Bollobás, B. and Riordan, O. (2006) Percolation, Cambridge University Press.CrossRefGoogle Scholar
[3]Grimmett, G. (1999) Percolation, 2nd edition, Springer.CrossRefGoogle Scholar
[4]Hammersley, J. M. and Welsh, D. J. A. (1965) First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Bernoulli, Bayes, Laplace: Anniversary Volume, Springer, pp. 61–110.Google Scholar
[5]Howard, C. D. and Newman, C. M. (2001) Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Probab. 29 577–623.CrossRefGoogle Scholar
[6]Johansson, K. (2000) Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.CrossRefGoogle Scholar
[7]Kardar, M., Parisi, G. and Zhang, Y.-C. (1986) Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889–892.CrossRefGoogle ScholarPubMed
[8]Kesten, H. (1993) On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296–338.CrossRefGoogle Scholar
[9]Kingman, J. F. C. (1968) The ergodic theory. J. Roy. Statist. Soc. Ser. B 30 499–510.Google Scholar
[10]Ligget, T. M., Schonmann, R. H. and Stacey, A. M. (1997) Domination by product measures. Ann. Probab. 25 71–95.Google Scholar
[11]Pimentel, L. P. R. (2004) Competing growth, interfaces and geodesics in first-passage percolation on Voronoi tilings. PhD thesis, IMPA, Rio de Janeiro.Google Scholar
[12]Pimentel, L. P. R. (2006) The time constant and critical probabilities in percolation models. Elect. Comm. Probab. 11 160–167.Google Scholar
[13]Pimentel, L. P. R. (2010) On Some fundamental aspects of Polyominoes on Random Voronoi Tilings. Preprint, available at: http//arxiv.org/abs/1009.3898v1.Google Scholar
[14]Vahidi-Asl, M. Q. and Wierman, J. C. (1990) First-passage percolation on the Voronoi tessellation and Delaunay triangulation. In Random Graphs 87 (Karonske, M., Jaworski, J. and Rucinski, A., eds), pp. 341–359.Google Scholar