Sharp phase transition for planarity of the Pareto soft random geometric graph

Determine whether the Pareto soft random geometric graph has a sharp phase transition for planarity, particularly in the regime where the Pareto exponent satisfies \(\alpha<\frac45d\).

Background

The paper establishes upper and lower bounds for the planarity threshold of the Pareto soft random geometric graph, whose edges exist when independently Pareto-weighted distances exceed the corresponding vertex distance. For $\alpha>\frac45d$, planarity is asymptotically governed by the appearance of $K_5$ subgraphs, whereas for smaller $\alpha$, non-planarity may arise from subgraphs whose order grows with the intensity parameter.

The authors explain that the thresholds obtained for $\alpha\leq\frac45d$ are not sharp because determining the existence of high-order non-planar subgraphs is difficult. They explicitly leave unresolved whether a sharp phase transition exists in this regime.