TY - JOUR
T1 - Percolation of finite clusters and infinite surfaces
AU - Grimmett, Geoffrey R.
AU - Holroyd, Alexander E.
AU - Kozma, Gady
N1 - Swiss National Science Foundation; EPSRC [EP/103372X/1]; Israel Science FoundationGRG acknowledges the hospitality of the Department of Mathematics at the University of British Columbia, the Section de Mathematiques at the University of Geneva and the Theory Group at Microsoft Research. He was supported in part by the Swiss National Science Foundation, and the EPSRC under grant EP/103372X/1. GK was supported in part by the Israel Science Foundation.
PY - 2014/3
Y1 - 2014/3
N2 - Two related issues are explored for bond percolation on ℤd (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 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.
AB - Two related issues are explored for bond percolation on ℤd (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 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.
UR - http://www.scopus.com/inward/record.url?scp=84897830487&partnerID=8YFLogxK
U2 - 10.1017/S030500411300073X
DO - 10.1017/S030500411300073X
M3 - مقالة
SN - 0305-0041
VL - 156
SP - 263
EP - 279
JO - Mathematical Proceedings of the Cambridge Philosophical Society
JF - Mathematical Proceedings of the Cambridge Philosophical Society
IS - 2
ER -