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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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