Lace Expansion and Mean-Field Behavior for the Random Connection Model

TL;DR

This paper develops a lace expansion for the continuum random connection model, establishing mean-field behavior and critical exponents in high dimensions or for spread-out/long-range connections.

Contribution

It introduces a lace expansion with a new BK inequality in the continuum and proves mean-field behavior for various connection functions in high dimensions.

Findings

Proves convergence of the lace expansion in high dimensions.

Establishes the triangle condition and infra-red bound.

Shows the critical exponent $b3=1$ and continuity of the percolation function.

Abstract

We study the random connection model driven by a stationary Poisson process. In the first part of the paper, we derive a lace expansion with remainder term in the continuum and bound the coefficients using a new version of the BK inequality. For our main results, we consider three versions of the connection function $\varphi$: a finite-variance version (including the Boolean model), a spread-out version, and a long-range version. For sufficiently large dimension (resp., spread-out parameter and $d>6$), we then prove the convergence of the lace expansion, derive the triangle condition, and establish an infra-red bound. From this, mean-field behavior of the model can be deduced. As an example, we show that the critical exponent $\gamma$ takes its mean-field value $\gamma=1$ and that the percolation function is continuous.