Percolation of finite clusters and infinite surfaces
Probability
2019-02-20 v2
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.
Keywords
Cite
@article{arxiv.1303.1657,
title = {Percolation of finite clusters and infinite surfaces},
author = {Geoffrey R. Grimmett and Alexander E. Holroyd and Gady Kozma},
journal= {arXiv preprint arXiv:1303.1657},
year = {2019}
}
Comments
v2: Minor revisions. To appear in the Mathematical Proceedings of the Cambridge Philosophical Society