Normal to Poisson phase transition for subgraph counting in the random-connection model

Abstract

We consider the limiting behavior of the count of subgraphs isomorphic to a graph $G$ with $m\ge 0$ fixed endpoints (or roots) in the random-connection model, as the intensity $\hat{I}»$ of the underlying Poisson point process tends to infinity. When connection probabilities are of order $\hat{I}{»}^{-\hat{I}\pm }$ we identify a phase transition phenomenon depending on a critical decay rate $\hat{I}{\pm }_m^{\ast }(G)>0$ such that normal approximation for subgraph counts holds when $\hat{I}\pm \in (0,\hat{I}{\pm }_m^{\ast }(G))$, and a Poisson limit result holds if $\hat{I}\pm =\hat{I}{\pm }_m^{\ast }(G)$. Our approach relies on cumulant growth rates derived by the convex analysis of planar diagrams that enumerate the partitions involved in cumulant identities. As a result, by the cumulant method we obtain normal approximation results with convergence rates in the Kolmogorov distance, and a Poisson limit theorem, for subgraph counts.