Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T19:09:33.259Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on renewal theory for percolation processes

Published online by Cambridge University Press:  14 July 2016

R. T. Smythe
R. T. Smythe*
Affiliation:
University of Washington, Seattle
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.

Keywords

FIRST-PASSAGE PERCOLATIONRENEWAL THEORYSUBADDITIVE PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 2 , June 1976 , pp. 290 - 300
Copyright
Copyright © Applied Probability Trust 1976 

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] Hammersley, J. M. (1966) First-passage percolation. J. R. Statist. Soc. B 28, 491–496.Google Scholar
[2] Hammersley, J. M. and Welsh, D. J. A. (1965) First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory, Bernoulli, Bayes, Laplace Anniversary Volume. Springer-Verlag, Berlin, 61–110.Google Scholar
[3] Kingman, J. F. C. (1968) The ergodic theory of subadditive stochastic processes. J. R. Statist. Soc. B 30, 499–510.Google Scholar
[4] Meyer, P. A. (1966) Probability and Potentials. Blaisdell, New York.Google Scholar