The Bauer–Golinelli formula for the atomic mass at zero

Let $G_n$ be a sparse Erdős–Rényi random graph whose limiting average degree is $cin(0,\infty)$, and let $\mu_c$ denote the limiting spectral measure of its adjacency matrix. For $q(c)$, let $q(c)$ be the smallest point $q\in(0,1)$ satisfying

$$q=e^{-ce^{-cq}}.$$

Bauer–Golinelli conjecture. For any $c\in(0,\infty)$,

$$\mu_c(\{0\})=q(c)+e^{-cq(c)}+cq(c)e^{-cq(c)}-1.$$

This predicts the limiting fraction of zero eigenvalues, or equivalently the asymptotic normalized nullity, in the sparse Erdős–Rényi regime. The formula was obtained by Bauer and Golinelli using the replica-symmetric ansatz; no resolution status is supplied in the source.