Percolation of finite clusters and infinite surfaces

TL;DR

This paper investigates the conditions under which infinite clusters and surfaces emerge in bond percolation on high-dimensional cubic lattices, establishing bounds and inequalities for critical probabilities related to these phenomena.

Contribution

It introduces new bounds for the critical probability where the complement of the infinite cluster contains an infinite component and for the existence of infinite dual surfaces, especially in high dimensions.

Findings

p_fin >= p_c, with strict inequality for large d or sufficiently spread-out models

p_surf >= p_fin, indicating a hierarchy of critical probabilities

Conditions under which infinite clusters and surfaces exist in high-dimensional lattices

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.