Convergence of the expected number of \(K_5\) subgraphs at the critical planarity scale

Determine whether the expected number of \(K_5\) subgraphs in the Pareto soft random geometric graph converges, rather than merely remaining within matching asymptotic upper and lower bounds, when \(t r_t^{4d/5}\) converges to a positive finite constant.

Background

For the Pareto soft random geometric graph, the planarity analysis considers the critical regime in which $t r_t^{4d/5}\to\lambda\in(0,\infty)$. In this regime, the expected number of $K_5$ subgraphs is bounded above and below at the same asymptotic order, and the limiting planarity probability would be $e^{-\gamma}$ if that expectation converged to a finite positive limit $\gamma$.

The authors explicitly state that their estimates do not establish convergence of the expectation itself; it may instead fluctuate within the common order of the upper and lower bounds. Thus, identifying whether the expectation has a limit at the critical scale remains unresolved.