Bond percolation on subsets of the square lattice, and the threshold between one-dimensional and two-dimensional behaviour

Abstract

For each value of pi in (1/2,1), the author constructs a subgraph R( pi ) of the square lattice Z2 such that the bond percolation process on R( pi ) has critical probability pi . It is shown that the critical probability pi (a) of the region ((x,y):0<or=y<or=a ln(x+1), 0<or=x< infinity ) depends in a non-trivial way upon the choice of the number a; pi (a) is a continuous, decreasing function of a and satisfies pi (a) to 1/2 as a to infinity and pi (a) to 1 as a down arrow 0. A family of 'one-dimensional' bond percolation processes is also constructed, whose critical probabilities range over the interval (0,1). All proofs are rigorous.