The direct-connectedness function in the random connection model

We investigate expansions for connectedness functions in the random connection model of continuum percolation powers intensity. Precisely, we study pair-connectedness and direct-connectedness functions, related to each other via Ornstein-Zernike equation. exhibit fact that coefficients consist sums over connected $2$-connected graphs. In physics literature, this is known be case more generally models based on Gibbs point processes stands analogy formalism developed correlation liquid-state statistical mechanics. find a representation function bounds intensity which allow us pass thermodynamic limit. some cases (e.g., high dimensions), results are valid almost entire subcritical regime. Moreover, relate these literature show how they coincide with expression provided by lace expansion.