The random connection model in high dimensions

By Ronald Meester, Mathew D. Penrose, and Anish Sarkar

Consider a continuum percolation model in which each pair of points of a $d$-dimensional Poisson process of intensity $\lambda$ is connected with a probability which is a function $g$ of the distance between them. We show that under a mild regularity condition on $g$, the critical value of $\lambda$, above which an infinite cluster exists almost surely, is asymptotic, as $d$ goes to infinity, to the inverse of the integral of $g(|x|)$ over $d$-dimensional space $R^d$.

Statistics and Probability Letters 35, 145-153 (1997).