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Nearest neighbor and hard sphere models in continuum percolation
Author(s) -
Häggström Olle,
Meester Ronald
Publication year - 1996
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/(sici)1098-2418(199610)9:3<295::aid-rsa3>3.0.co;2-s
Subject(s) - continuum percolation theory , percolation threshold , tangent , percolation (cognitive psychology) , percolation critical exponents , directed percolation , poisson point process , ball (mathematics) , statistical physics , poisson distribution , mathematics , combinatorics , k nearest neighbors algorithm , dimension (graph theory) , physics , mathematical analysis , geometry , quantum mechanics , computer science , statistics , neuroscience , artificial intelligence , biology , electrical resistivity and conductivity
Consider a Poisson process X in R d with density 1. We connect each point of X to its k nearest neighbors by undirected edges. The number k is the parameter in this model. We show that, for k = 1, no percolation occurs in any dimension, while, for k = 2, percolation occurs when the dimension is sufficiently large. We also show that if percolation occurs, then there is exactly one infinite cluster. Another percolation model is obtained by putting balls of radius zero around each point of X and let the radii grow linearly in time until they hit another ball. We show that this model exists and that there is no percolation in the limiting configuration. Finally we discuss some general properties of percolation models where balls placed at Poisson points are not allowed to overlap (but are allowed to be tangent). © 1996 John Wiley & Sons, Inc.