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On the time constant in a dependent first passage percolation model

Published online by Cambridge University Press:  27 March 2014

Julie Scholler
Julie Scholler*
Affiliation:
Universitéde Lorraine, Institut Élie Cartan de Lorraine, UMR 7502, 54506 Vandoeuvre-lès-Nancy, France. [email protected]
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Abstract

We pursue the study of a random coloring first passage percolation model introduced byFontes and Newman. We prove that the asymptotic shape of this first passage percolationmodel continuously depends on the law of the coloring. The proof uses several couplings,particularly with greedy lattice animals.

Keywords

First passage percolationpercolationtime constantrandom coloring
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 171 - 184
Copyright
© EDP Sciences, SMAI, 2014

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References

P. Billingsley, Probability and Measure. Wiley-Interscience (1995).
Boivin, D., First passage percolation: the stationary case. Probab. Theory Related Fields 86 (1990) 491–499. Google Scholar
Cox, J.T., The time constant of first-passage percolation on the square lattice. Adv. Appl. Probab. 12 (1980) 864–879. Google Scholar
Cox, J.T., Gandolfi, A., Griffin, P.S. and Kesten, H., Greedy lattice animals. I. Upper bounds. Ann. Appl. Probab. 3 (1993) 1151–1169. Google Scholar
Cox, J.T. and Kesten, H., On the continuity of the time constant of first-passage percolation. J. Appl. Probab. 18 (1981) 809–819. Google Scholar
Fontes, L. and van Newman, C.M., First passage percolation for random colorings of Zd. Ann. Appl. Probab. 3 (1993) 746–762. Google Scholar
Gandolfi, A. and Kesten, H., Greedy lattice animals. II. Linear growth. Ann. Appl. Probab. 4 (1994) 76–107. Google Scholar
G. Grimmett, Percolation, in vol. 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], second edition. Springer-Verlag, Berlin (1999).
J.M. Hammersley and D.J.A. Welsh, First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory, in Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif. Springer-Verlag, New York (1965) 61–110.
C.D. Howard, Models of first-passage percolation, in Probability on discrete structures, vol. 110 of Encyclopaedia Math. Sci. Springer, Berlin (2004) 125–173.
Kesten, H., Analyticity properties and power law estimates of functions in percolation theory. J. Stat. Phys. 25 (1981) 717–756. Google Scholar
H. Kesten, Aspects of first passage percolation. In École d’été de probabilités de Saint-Flour, XIV—1984, vol. 1180 of Lect. Notes in Math. Springer, Berlin (1986) 125–264.
H. Kesten, First-passage percolation. From classical to modern probability, in vol. 54 of Progr. Probab. Birkhäuser, Basel (2003) 93–143.
Kingman, J.F.C., The ergodic theory of subadditive stochastic processes. J. Roy. Stat. Soc. Ser. B 30 (1968) 499–510. Google Scholar
Liggett, T.M., An improved subadditive ergodic theorem. Ann. Probab. 13 (1985) 1279–1285. Google Scholar
Martin, J.B., Linear growth for greedy lattice animals. Stoch. Process. Appl. 98 (2002) 43–66. Google Scholar