Summary
If the passage time of the edges of the ℤd lattice are stationary, ergodic and have finite moment of orderp>d, then a.s. the set of vertices that can be reached within timet, has an asymptotic shape ast→∞.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Boivin, D., Derriennic, Y.: The ergodic theorem for additive cocycles of ℤd and ℝ;d. Ergodic Theory Dyn. Syst. (to appear)
Broise, M.: Deniel, Y.: Deriennic, Y.: Réarrangements, inégalités maximales et théorèmes ergodiques fractionnaires. Ann. Inst. Fourier39, 1–26 (1989)
Calderon, A.: Ergodic theory and translation invariant operators. Proc. Natl. Acad. Sci. USA59, 349–358 (1968)
Cox, J.T., Durrett, R.: Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab.9, 583–603 (1981)
Durrett, R.: Lecture notes on particle systems and percolation. Grove, Calif. Wadsworth pacific 1988
Hammersley, J.M., Welsh, D.J.A.: First-passage percolation, subadditive processes, stochastic networks and generalized renwal theory. In: Bernoulli, Bayes, Laplace anniversary volume. Neyman, J., Le Cam, L.M. eds. pp. 61–110 Berlin Heidelberg New York: Springer 1965
Kesten, H.: Aspects of first-passage percolation. In: Hennequin, P.L. (ed.) Ecole d' Eté de Probabilités de Saint Flour XIV (Lect. Notes Math., vol. 1180, pp 126–265) Ecole de st. Flour, 1984 Berlin Heidelberg New York: Springer 1976
Kingman, J.F.C.: The ergodic theory of subadditive stochastic processes. J.R. Stat. Soc., Ser. B30, 499–510 (1968)
Krengel, U.: Ergodic theorems. Berlin New York: de Gruyter, 1985
Lorentz, G.G.: Some new functional spaces. Ann. Math.51, 37–55 (1950)
Ornstein, D., Weiss, B.: Ergodic theory of amenable group actions. I. The Rohlin lemma. Bull. Am. Math. Soc.2, 161–164 (1980)
Smythe, R.T.: Wierman, J.C.: First-passage percolation on the square lattice. (Lect. Notes Math., vol. 671) Berlin Heidelberg New York: Springer 1978
Stein, E.M., Weiss, G.: Introduction to Fourier analysis on euclidean spaces. Princeton: Princeton University Press 1971
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Boivin, D. First passage percolation: The stationary case. Probab. Th. Rel. Fields 86, 491–499 (1990). https://doi.org/10.1007/BF01198171
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01198171