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arXiv · 1303.1657

Percolation of finite clusters and infinite surfaces

Geoffrey R. Grimmett·Alexander E. Holroyd·Gady Kozma
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Abstract

Two related issues are explored for bond percolation on the d-dimensional cubic lattice (with d > 2) 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 p_fin satisfies p_fin >= p_c, 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 p_surf satisfies p_surf >= p_fin.

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Geoffrey R. Grimmett, Alexander E. Holroyd, Gady Kozma. 2013-10-16. Percolation of finite clusters and infinite surfaces. https://doi.org/10.1017/s030500411300073x

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