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Weak moment conditions for time coordinates in first-passage percolation models

Published online by Cambridge University Press:  14 July 2016

John C. Wierman
John C. Wierman*
Affiliation:
University of Minnesota
*
∗Postal address: School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455, U.S.A.
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Abstract

A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.

Keywords

FIRST-PASSAGE PERCOLATIONSUBADDITIVE PROCESSGENERALIZED RENEWAL THEORYTIME CONSTANTHEIGHT PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 4 , December 1980 , pp. 968 - 978
Copyright
Copyright © Applied Probability Trust 

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Footnotes

Research supported by NSF MCS 78–001168.

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. Bernoulli–Bayes–Laplace Anniversary Volume, Springer-Verlag, Berlin, 61–110.Google Scholar
Kingman, J. F. C. (1973) Subadditive ergodic theory. Ann. Prob. 1, 883–909.Google Scholar
Reh, W. (1979) First-passage percolation under weak moment conditions. J. Appl. Prob. 16, 750–763.Google Scholar
Smythe, R. T. (1976) Remarks on renewal theory for percolation processes. J. Appl. Prob. 13, 290–300.Google 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 Wierman, J. C. (1978b) First-Passage Percolation on the Square Lattice. Lecture Notes in Mathematics 671, 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