Optimal second eigenvalue bound above a critical dimension
Establish that, above a critical dimension, the second-largest eigenvalue (in absolute value) of the adjacency matrix of G(n,d,p) achieves the optimal order O(√(np)), eliminating the current polylogarithmic factors in available bounds.
References
We conjecture that, above a critical dimension, the second largest eigenvalue of $\mathcal G(n,d,p)$ indeed satisfies the optimal $O(\sqrt{np})$ bound.
— Spectra of high-dimensional sparse random geometric graphs
(Cao et al., 9 Jul 2025) in Introduction, Spectral gap of random graphs