Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T22:30:00.779Z Has data issue: false hasContentIssue false
  • English
  • Français

First-passage percolation under weak moment conditions

Published online by Cambridge University Press:  14 July 2016

Wolfgang Reh
Wolfgang Reh*
Affiliation:
Universität Mannheim
Get access Rights & Permissions [Opens in a new window]

Abstract

Most research in first-passage percolation has been done under the assumption of a finite mean for the underlying time coordinate distribution. We demonstrate that the basic ergodic results can be derived under a weaker moment assumption, which still permits us to evaluate the time constant in the case where the atom at zero of the time coordinate distribution exceeds one-half. Further almost sure convergence is investigated more closely.

Keywords

FIRST-PASSAGE PERCOLATIONSUBADDITIVE PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 4 , December 1979 , pp. 750 - 763
Copyright
Copyright © Applied Probability Trust 1979 

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

Hammersley, J. M. (1966) First-passage percolation. J. R. Statist. Soc. B 28, 491–496.Google Scholar
Hammersley, J. M. (1974) Postulates for subadditive processes. Ann. Prob. 2, 652–680.Google Scholar
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–Verlag, Berlin, 61–110.Google Scholar
Kingman, J. F. C. (1968) The ergodic theory of subadditive processes. J. R. Statist. Soc. B 30, 499–510.Google Scholar
Kingman, J. F. C. (1973) Subadditive ergodic theory. Ann. Prob. 1, 883–909.Google Scholar
Seymour, P. D. and Welsh, D. J. A. (1978) Percolation probabilities on the square lattice. Ann. Discrete Math. 3, 227–245.CrossRefGoogle Scholar
Smythe, R. T. (1976) Remarks on renewal theory for percolation processes. J. Appl. Prob. 13, 290–300.CrossRefGoogle Scholar
Smythe, R. T. and Wierman, J. C. (1977) First-passage percolation on the square lattice, I. Adv. Appl. Prob 9, 38–54.Google Scholar
Smythe, R. T. and Wierman, J. C. (1978a) First-passage percolation on the square lattice, III. Adv. Appl. Prob. 10, 155–171.Google Scholar
Smythe, R. T. and Wiermann, J. C. (1978b) First-passage Percolation of the Square Lattice. Lecture Notes in Mathematics, Springer–Verlag, Berlin.Google Scholar
Wierman, J. C. (1977) First-passage percolation on the square lattice, II. Adv. Appl. Prob. 9, 283–295.Google Scholar
Wierman, J. C. and Reh, W. (1978) On conjectures in first passage percolation theory. Ann. Prob. 6, 388–397.Google Scholar