The Correlation Length for the High-Density Phase of Bernoulli Percolation

Abstract

We examine two standard types of connectivity functions in the high-density phase of nearest-neighbor Bernoulli (bond) percolation. We show that these two quantities decay exponentially at the same constant rate. The reciprocal of this constant defines therefore a correlation length. Unfortunately, we cannot prove that this correlation length is finite whenever $p > p_c$, although previous work established this result for $p$ above a threshold which is conjectured to coincide with $p_c$. We examine also a third connectivity function and prove that it too decays exponentially with the same rate as the two standard connectivity functions. We establish various useful properties of our correlation length, such a semicontinuity as a function of bond density and convexity in its directional dependence. Finally, for bond percolation in two dimensions we show that the correlation length at bond density $p_1 > p_c = \frac{1}{2}$ is exactly half the correlation length at the subcritical bond density $p_2 = 1 - p_1 < p_c$. This sharpens some other exact results for two-dimensional percolation and is the precise analog of known results for the two-dimensional Ising model.