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Long range percolation in stationary point processes
Author(s) -
Burton Robert M.,
Meester Ronald W. J.
Publication year - 1993
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/rsa.3240040205
Subject(s) - uniqueness , percolation (cognitive psychology) , mathematics , statistical physics , point process , directed percolation , range (aeronautics) , phase transition , percolation theory , continuum percolation theory , combinatorics , mathematical analysis , percolation critical exponents , physics , statistics , topology (electrical circuits) , critical exponent , quantum mechanics , neuroscience , materials science , composite material , biology
We consider a continuum percolation model in ℝ d , d ⩾ 1 in which any two points of a stationary point process are connected with a probability which decays exponentially in the distance between the points. We give sufficient conditions for the (non)‐existence of a phase transition. We also give examples of processes which show that it is impossible to write down a theorem which relates the critical parameter value of a process to its density. Finally, we show that uniqueness of the infinite cluster is still valid in this general setting. © 1993 John Wiley & Sons, Inc.
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