Closed-form density for the limiting distribution ν_α
Derive a closed-form expression for the density of the limiting spectral distribution ν_α that arises for sparse graphs with expected degree α and governs the limit for G(n,d,α/n).
References
Moreover, as $\alpha\to\infty$, $\nu_{\alpha}$ converges back to the semicircle law, although no closedâform density is known for $\nu_{\alpha}$.
— Spectra of high-dimensional sparse random geometric graphs
(Cao et al., 9 Jul 2025) in Section 2.2, Limiting spectral distribution for $\mathcal G(n,d,\frac{\alpha}{n})$