Optimal O(√(np)) bound for the second eigenvalue above a critical dimension

Prove that for the high-dimensional random geometric graph G(n,d,p) on the sphere with adjacency matrix A, there exists a dimension threshold (as a function of n and p) such that whenever d exceeds this critical threshold, the second-largest eigenvalue in absolute value satisfies λ(A) = O(√(np)) with high probability.

Background

The authors obtain a near-optimal bound λ(A) = O(log4(n)√(np)) when d = Ω(np), thereby matching Erdős–Rényi expansion up to a polylogarithmic factor.

They posit that an optimal O(√(np)) bound should hold once the ambient dimension is sufficiently large, but establishing this would likely require new combinatorial counting techniques beyond the classical high-trace method.

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 (2507.06556 - Cao et al., 9 Jul 2025) in Introduction, subsection “Spectral gap of random graphs”