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

The Direct-Connectedness Function in the Random Connection Model