Advances in Applied Probability (1)
Mathematical Proceedings of the Cambridge Philosophical Society (1)
Proceedings of the London Mathematical Society (1)
Applied Probability Trust (1)
Influence in product spaces
Foundations of probability theory
Classical measure theory
Geoffrey R. Grimmett, Svante Janson, James R. Norris
Journal: Advances in Applied Probability / Volume 48 / Issue A / July 2016
The theory of influence and sharp threshold is a key tool in probability and probabilistic combinatorics, with numerous applications. One significant aspect of the theory is directed at identifying the level of generality of the product probability space that accommodates the event under study. We derive the influence inequality for a completely general product space, by establishing a relationship to the Lebesgue cube studied by Bourgain, Kahn, Kalai, Katznelson and Linial (BKKKL) in 1992. This resolves one of the assertions of BKKKL. Our conclusion is valid also in the setting of the generalized influences of Keller.
Percolation of finite clusters and infinite surfaces
GEOFFREY R. GRIMMETT, ALEXANDER E. HOLROYD, GADY KOZMA
Journal: Mathematical Proceedings of the Cambridge Philosophical Society / Volume 156 / Issue 2 / March 2014
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.
ENTANGLEMENT IN PERCOLATION
GEOFFREY R. GRIMMETT, ALEXANDER E. HOLROYD
Journal: Proceedings of the London Mathematical Society / Volume 81 / Issue 2 / September 2000
We study finite and infinite entangled graphs in the bond percolation process in three dimensions with density $p$. After a discussion of the relevant definitions, the entanglement critical probabilities are defined. The size of the maximal entangled graph at the origin is studied for small $p$, and it is shown that this graph has radius whose tail decays at least as fast as $\exp(-\alpha n/\log n)$; indeed, the logarithm may be replaced by any iterate of logarithm for an appropriate positive constant $\alpha$. We explore the question of almost sure uniqueness of the infinite maximal open entangled graph when $p$ is large, and we establish two relevant theorems. We make several conjectures concerning the properties of entangled graphs in percolation. http://www.statslab.cam.ac.uk/$\sim$grg/ 1991 Mathematics Subject Classification: primary 60K35; secondary 05C10, 57M25, 82B41, 82B43, 82D60.