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On the continuity of the time constant of first-passage percolation

Published online by Cambridge University Press:  14 July 2016

J. Theodore Cox  and
Harry Kesten
J. Theodore Cox*
Affiliation:
Syracuse University
Harry Kesten*
Affiliation:
Cornell University
*
∗Postal address: Department of Mathematics, Syracuse University, 200 Carnegie, Syracuse, NY 13210, U.S.A.
∗∗Postal address: Department of Mathematics, White Hall, Cornell University, Ithaca, NY 14850, U.S.A.
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Abstract

Let U be the distribution function of the non-negative passage time of an individual edge of the square lattice, and let a0n be the minimal passage time from (0, 0) to (n, 0). The process a0n/n converges in probability as n → ∞to a finite constant μ (U) called the time constant. It is proven that μ (Uk)→ μ(U) whenever Uk converges weakly to U.

Keywords

FIRST-PASSAGE PERCOLATIONTIME CONSTANT
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 4 , December 1981 , pp. 809 - 819
Copyright
Copyright © Applied Probability Trust 1981 

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References

[1] Brånvall, G. (1979) Some results in percolation theory. Uppsala University, Department of Mathematics Report No. 11.Google Scholar
[2] Cox, J. T. (1980) The time constant of first-passage percolation on the square lattice. Adv. Appl. Prob. 12, 864–879.CrossRefGoogle Scholar
[3] Cox, J. T. and Durrett, R. (1980) Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Prob. 8.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 , ed. Neyman, J. and LeCam, L., Springer-Verlag, Berlin.Google Scholar
[5] Janson, S. (1981) An upper bound for the velocity of first-passage percolation. J. Appl. Prob. 18, 256–262.CrossRefGoogle Scholar
[6] Kesten, H. (1980) On the time constant and path length of first-passage percolation. Adv. Appl. Prob. 12, 848–863.CrossRefGoogle Scholar
[7] Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals ½ Commun. Math. Phys. 74, 41–59.CrossRefGoogle Scholar
[8] Kingman, J. F. C. (1968) The ergodic theory of subadditive processes. J. R. Statist. Soc. B 30, 499–510.Google Scholar
[9] Reh, W. (1979) First-passage percolation under weak moment conditions. J. Appl. Prob. 16, 750–763.CrossRefGoogle Scholar
[10] Runnels, L. K. and Lebowitz, J. L. (1976) Analyticity of a hard-core multicomponent lattice gas. J. Statist. Phys. 14, 525–533.CrossRefGoogle Scholar
[11] Schürger, K. (1980) On the asymptotic behavior of percolation processes. J. Appl. Prob. 17, 385–402.CrossRefGoogle Scholar
[12] Smythe, R. and Wierman, J. C. (1978) First Passage Percolation on the Square Lattice. Lecture Notes in Mathematics 671, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[13] Wierman, J. C. (1980) Weak moment conditions for time coordinates in first-passage percolation. J. Appl. Prob. 17, 968–978.CrossRefGoogle Scholar
[14] Wierman, J. C. and Reh, W. (1979) On conjectures in first-passage percolation. Ann. Prob. 6, 388–397.Google Scholar