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On the asymptotic geometrical behaviour of percolation processes

Published online by Cambridge University Press:  14 July 2016

Klaus Schürger
Klaus Schürger*
Affiliation:
Deutsches Krebsforschungszentrum, Heidelberg
*
∗Postal address: Deutsches Krebsforschungszentrum, Institut für Dokumentation, Information und Statistik, Im Neuenheimer Feld 280, D–6900 Heidelberg, W. Germany.
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Abstract

In this paper the global behaviour of percolation processes on the d-dimensional square lattice is studied. Using techniques of Richardson (1973) we prove, under weak moment assumptions on the time coordinate distribution, the following result. There exists a norm N(·) on Rd such that, for all 0 < ε < 1, we have that almost surely for all sufficiently large t the N-ball of radius (1 – ε)t is contained in η, (the set of all sites occupied by time t) and η, is contained in the N-ball of radius (1 + ε)t. Richardson (1973) derived the corresponding ‘in probability' result for a class of spread processes on Rd, satisfying certain conditions.

Keywords

FIRST-PASSAGE PERCOLATIONSUBADDITIVE PROCESSESSPATIAL PROCESSESSTRONG LAW OF LARGE NUMBERSLIMIT THEOREMSPOPULATION GROWTHINFECTION MODELSGEOMETRICAL PROPERTIES OF PERCOLATION PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 2 , June 1980 , pp. 385 - 402
Copyright
Copyright © Applied Probability Trust 1980 

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References

Baum, L. E. and Katz, M. (1965) Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, 108–123.Google Scholar
Biggins, J. D. (1978) The asymptotic shape of the branching random walk. Adv. Appl. Prob. 10, 62–84.Google Scholar
Hammersley, J. M. (1957a) Percolation processes. The connective constant. Proc. Camb. Phil. Soc. 53, 642–645.Google Scholar
Hammersley, J. M. (1957b) Percolation processes. Lower bounds for the critical probability. Ann. Math. Statist. 28, 790–795.Google Scholar
Hammersley, J. M. (1961) On the rate of convergence to the connective constant of the hypercubical lattice. Quart. J. Math. (Oxford) (2) 12, 250–256.Google Scholar
Hammersley, J. M. (1962) Generalization of the fundamental theorem on subadditive functions. Proc. Camb. Phil. Soc. 58, 235–238.Google Scholar
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. (1962) Further results on the rate of convergence to the connective constant of the hypercubical lattice. Quart. J. Math. (Oxford) (2) 13, 108–110.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
Harris, T. E. (1974) Contact interactions on a lattice. Ann. Prob. 2, 969–988.Google Scholar
Kesten, H. (1973) Contribution to the discussion in Kingman (1973), p. 903.Google Scholar
Kingman, J. F. C. (1968) The ergodic theory of subadditive stochastic processes. J. R. Statist. Soc. B 30, 499–510.Google Scholar
Kingman, J. F. C. (1973) Subadditive ergodic theory. Ann. Prob. 1, 883–909.Google Scholar
Kingman, J. F. C. (1976) Subadditive Processes. Lecture Notes in Mathematics 539, Springer-Verlag, Berlin, 167–223.Google Scholar
Mollison, D. (1972) Conjecture on the spread of infection in two dimensions disproved. Nature 240, 467–468.Google Scholar
Mollison, D. (1978) Markovian contact processes. Adv. Appl. Prob. 10, 85–108.Google Scholar
Reh, W. (1979) First-passage percolation under weak moment conditions. J. Appl. Prob. 16, 750–763.Google Scholar
Richardson, D. (1973) Random growth in a tessellation. Proc. Camb. Phil. Soc. 74, 515–528.Google Scholar
Schürger, K. (1979) On the asymptotic geometrical behaviour of a class of contact interaction processes with a monotone infection rate. Z. Wahrscheinlichkeitsth. 48, 35–48.Google Scholar
Schürger, K. and Tautu, P. (1976) A Markovian configuration model for carcinogenesis. In Mathematical Models in Medicine, ed. Berger, J. et al. Lecture Notes in Biomathematics 11, Springer-Verlag, Berlin, 92–108.Google Scholar
Seymour, P. D. and Welsh, D. J. A. (1978) Percolation probabilities on the square lattice. In Advances in Graph Theory, ed. Bollobás, B. North-Holland, Amsterdam, 227–245.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. (1978) 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