The spectral conjecture for detection in random geometric graphs

Let $obreak\boldsymbol{x}_1, obreak\ldots, obreak\boldsymbol{x}_n$ be latent points on the sphere, and let

$$be the normalized kernel operator on the sphere associated with the random geometric graph (RGG) kernel. Write$$

for the trace of the cube of this operator.

Spectral detection conjecture. The sharp threshold for distinguishing the RGG from its Erdős–Rényi counterpart is determined by

$$n^3\operatorname{tr}^2(\kappa^3)=1.$$

Equivalently, the detectable regime should satisfy $n^3\operatorname{tr}^2(\kappa^3)\gg1$, while the undetectable regime should satisfy $n^3\operatorname{tr}^2(\kappa^3)\ll1$.

The conjecture is motivated by both the signed triangle statistic and the Kullback–Leibler expansion, where $\operatorname{tr}(\kappa^3)$ arises naturally. It unifies the conjectured threshold for hard RGGs with the proved thresholds for several smooth-kernel models, but the general sharp characterization remains open.