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Percolation of finite clusters and infinite surfaces

Published online by Cambridge University Press:  15 January 2014

Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB. e-mail:, URL:
Microsoft Research, 1 Microsoft Way, Redmond, WA 98052, U.S.A. e-mail:, URL:
Department of Mathematics, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel. e-mail:, URL:


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 pfinpc, 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 psurfpfin.

Research Article
Copyright © Cambridge Philosophical Society 2014 

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