The Direct-Connectedness Function in the Random Connection Model

TL;DR

This paper analyzes the expansions of connectedness functions in the continuum percolation model, linking graph sums to physical formalisms, and extends results to the thermodynamic limit, especially in high dimensions.

Contribution

It provides a detailed expansion of the direct-connectedness function in the random connection model, connecting graph sums with physical correlation functions and establishing bounds for the thermodynamic limit.

Findings

Coefficients involve sums over connected and 2-connected graphs.

Representation and bounds enable passage to the thermodynamic limit.

Results applicable in high dimensions within the subcritical regime.

Abstract

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