Born–Green hierarchy for continuum percolation

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Abstract
We present a projection operator technique that yields hierarchies of integral equations satisfied exactly by the n-point connectedness functions in a continuum version of the site-bond percolation problem. The n-point connectedness functions carry the same structural information for a percolation problem as the n-point correlation functions do for a thermal problem. Our method extends the Potts model mapping of Fortuin and Kastelyn to the continuum. We use the projection operator technique to produce an integral equation hierarchy for percolation similar to the Born–Green thermal hierarchy. The Kirkwood superposition approximation (SA) is extended to percolation in order to close this hierarchy and yield a nonlinear integral equation for the two-point connectedness function. We discuss the fact that this function, in the SA, is the analytic continuation to negative density of the two-point correlation function in a corresponding thermal problem. The Born–Green–Yvon (BGY) equation for percolation is solved numerically, both by an expansion in powers of the density, and iteratively, using the modified Picard method. We argue, both analytically and numerically, that the BGY equation for percolation, unlike its thermal counterpart, shows nonclassical critical behavior, with η=1 and γ=2.2±0.2. Finally, we develop a sequence of refinements to the superposition approximation that can be used to give increasingly accurate calculations of the two-point connectedness function.

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