Threshold for semicircle law in sparse high-dimensional random geometric graphs
Determine the exact dimension threshold for semicircle convergence in the high-dimensional random geometric graph G(n,d,p); specifically, prove that the empirical spectral distribution of A/√(np(1−p)) converges to the semicircle law under the condition d ≫ np log(1/p), thereby removing the extra logarithmic factor from the current assumption d = ω(np log^2(1/p)).
References
Based on the entropic and spectral thresholds $d\asymp np\log(1/p)$ pointed out in , we conjecture that the true threshold for semicircle convergence is in fact $d\gg np\log(1/p)$, removing just a single logarithmic factor from our current condition.
— Spectra of high-dimensional sparse random geometric graphs
(Cao et al., 9 Jul 2025) in Introduction, Testing high-dimensional geometry