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Critical Probabilities of 1-Independent Percolation Models

Published online by Cambridge University Press:  02 February 2012

PAUL BALISTER  and
BÉLA BOLLOBÁS
PAUL BALISTER
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA (e-mail: pbalistr@memphis.edu)
BÉLA BOLLOBÁS
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA (e-mail: pbalistr@memphis.edu) Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WA, UK (e-mail: b.bollobas@dpmms.cam.ac.uk)
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Abstract

Given a locally finite connected infinite graph G, let the interval [pmin(G), pmax(G)] be the smallest interval such that if p > pmax(G), then every 1-independent bond percolation model on G with bond probability p percolates, and for p < pmin(G) none does. We determine this interval for trees in terms of the branching number of the tree. We also give some general bounds for other graphs G, in particular for lattices.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 1-2: Honouring the Memory of Richard H. Schelp , March 2012 , pp. 11 - 22
Copyright
Copyright © Cambridge University Press 2012

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