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The time constant of first-passage percolation on the square lattice

Published online by Cambridge University Press:  01 July 2016

J. Theodore Cox
J. Theodore Cox*
Affiliation:
Syracuse University
*
∗Postal address: Department of Mathematics, 200 Carnegie, Syracuse University, Syracuse, NY 13210, U.S.A.
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Abstract

Let μ (F) be the time constant of first-passage percolation on the square lattice with underlying distribution function F. Two theorems are presented which show, under some restrictions, that μ varies continuously in F with respect to weak convergence. These results are improvements of existing continuity theorems.

Keywords

FIRST-PASSAGE PERCOLATIONTIME CONSTANTSUBADDITIVE PROCESSES

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 4 , December 1980 , pp. 864 - 879
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

Research supported in part by NSF Grant No. MCS76-82358.

References

[1] Badrikian, A., Kingman, J. F. C. and Kuelbs, J. (1976) Ecole d'Eté de Probabilités de Saint-Flour V-1975. Lecture Notes in Mathematics 539, Springer-Verlag, New York.Google Scholar
[2] Hammersley, J. M. (1966) First-passage percolation. J. R. Statist. Soc. B28, 491–496.Google Scholar
[3] 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 , ed. Neyman, J. and LeCam, L. M. Springer-Verlag, Berlin.Google Scholar
[4] Kesten, H. (1980) On the time constant and path length of first-passage percolation. Adv. Appl. Prob. 12, 848–863.Google Scholar
[5] Richardson, D. (1974) Random growth in a tessellation. Proc. Camb. Phil. Soc. 70, 515–528.Google Scholar
[6] Smythe, R. T. and Wierman, J. C. (1978) First-Passage Percolation on the Square Lattice. Lecture Notes in Mathematics 671, Springer-Verlag, New York.Google Scholar
[7] Wierman, J. C. (1980) Weak moment conditions for the time coordinates in first-passage percolation models. J. Appl. Prob. 17, 966–976.Google Scholar