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Percolation of finite clusters and infinite surfaces

Published online by Cambridge University Press:  15 January 2014

GEOFFREY R. GRIMMETT ,
ALEXANDER E. HOLROYD  and
GADY KOZMA
GEOFFREY R. GRIMMETT
Affiliation:
Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB. e-mail: g.r.grimmett@statslab.cam.ac.uk, URL: http://www.statslab.cam.ac.uk/~grg/
ALEXANDER E. HOLROYD
Affiliation:
Microsoft Research, 1 Microsoft Way, Redmond, WA 98052, U.S.A. e-mail: holroyd@microsoft.com, URL: http://research.microsoft.com/~holroyd/
GADY KOZMA
Affiliation:
Department of Mathematics, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel. e-mail: gady.kozma@weizmann.ac.il, URL: http://www.wisdom.weizmann.ac.il/~gadyk/
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Abstract

Two related issues are explored for bond percolation on ${\mathbb{Z}^d$ (with d ≥ 3) and its dual plaquette process. Firstly, for what values of the parameter p does the complement of the infinite open cluster possess an infinite component? The corresponding critical point pfin satisfies pfin ≥ pc, and strict inequality is proved when either d is sufficiently large, or d ≥ 7 and the model is sufficiently spread out. It is not known whether d ≥ 3 suffices. Secondly, for what p does there exist an infinite dual surface of plaquettes? The associated critical point psurf satisfies psurf ≥ pfin.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 2 , March 2014 , pp. 263 - 279
Copyright
Copyright © Cambridge Philosophical Society 2014 

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