On a conjecture of Hammersley and Whittington concerning bond percolation on subsets of the simple cubic lattice

Abstract

The author verifies the truth of a conjecture of Hammersley and Whittington (1985) concerning bond percolation on certain subsets of the simple cubic lattice Z3. Let f and g be non-decreasing, non-negative functions on (0, infinity ) and let Z3(f, g) denote the (f,g)-wedge of Z3, being the set of points (x, y, z) such that 0<or=y<or=f(x), O<or=z<or=g(x) and x>or=0. The author shows that the condition (1+f(x))(1+g(x)) to infinity as x to infinity is sufficient for the critical probability of the bond percolation process on Z3(f,g) to be less than or equal to 1/2.