Abstract: We show that the critical dimension for distinguishing a random geometric graph with a smooth kernel from its Erdos–Renyi counterpart is given by $d_* = n^{3/4}$, much lower than $d_* = n^3$ for the hard RGG in [Bubeck-Ding-Eldan-Racz 2016]. To unify these results, we formulate a conjecture that the critical dimension is spectrally determined by $d = n^{3/4} b_1^{3/2}$, where $b_1/d$ is the second eigenvalue of the kernel operator.