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

TL;DR

This paper studies the asymptotic distribution of subgraph counts in a random-connection model, revealing a phase transition from normal to Poisson limits depending on the decay rate of connection probabilities as the intensity grows.

Contribution

It identifies a critical decay rate dictating the transition from normal to Poisson limits for subgraph counts in the model, using cumulant analysis and convex planar diagram enumeration.

Findings

Normal approximation holds below the critical decay rate.

Poisson limit occurs at the critical decay rate.

Convergence rates in Kolmogorov distance are established.

Abstract

We consider the limiting behavior of the count of subgraphs isomorphic to a graph $G$ with $m\geq 0$ fixed endpoints (or roots) in the random-connection model, as the intensity $\lambda$ of the underlying Poisson point process tends to infinity. When connection probabilities are of order $\lambda^{-\alpha}$ we identify a phase transition phenomenon depending on a critical decay rate $\alpha^\ast_m (G)>0$ such that normal approximation for subgraph counts holds when $\alpha \in (0,\alpha^\ast_m (G) )$, and a Poisson limit result holds if $\alpha = \alpha^\ast_m (G)$. 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.