The random connection model in high dimensions
The random connection model is described as follows. Let \(X'= \{x_1,x_2, \dots\}\) be a homogeneous Poisson point process with intensity \(\lambda\) on \(\mathbb{R}^d\). The corresponding inhomogeneous process is \(X=\{x_0= 0\in\mathbb{R}^d\), \(x_1,x_2, \dots\}\). For \(g:(0, \infty] \to[0,1]\) a sufficiently quickly decreasing function, the points \(x_i\), \(x_j\in X\) are connected with probability \(g(| x_i- x_j|)\) independently on anything else. Now \(C\) is the cluster of the origin if \(C\) is the set of \(x_i\)'s that lie in the connected component of \(X\) that contains the origin. It is shown, roughly speaking, that, for \(d\) tending to infinity, the critical intensity \(\lambda_c^{(d)} (g)\) behaves like \(1/I_d (g)\) and the probability of the appearence of an infinite cluster of the origin behaves like \(1-\psi (\lambda I_d (g))\). Here \(\lambda_c^{(d)} (g)\) is the infimum of the intensities for which an infinite cluster of the origin appears with positive probability. Further, \(I_d(g)= \int_{\mathbb{R}^d} g(| x|) dx\) and \(\psi (\mu)\) is the extinction probability for a simple branching process with Poisson offspring distribution with mean \(\mu\).
scientific article; zbMATH DE number 1092052