The Bauer–Golinelli conjecture on emergence of extended states at zero

Let $\mu_c$ be the limiting spectral measure associated with the Poisson–Galton–Watson tree of mean offspring $c\in(0,\infty)$. A measure $\mu$ has no extended states at $E\in\mathbb{R}$ if

$$\frac{\mu([E-\varepsilon,E+\varepsilon])-\mu(\{E\})}{\varepsilon}\xrightarrow[\varepsilon\to0+]{}0,$$

and has extended states at $E$ otherwise.

Bauer–Golinelli conjecture. The following phase transition occurs: if $c<e$, then $\mu_c$ has no extended states at $0$; if $c>e$, then $\mu_c$ has extended states at $0$.

This conjecture predicts that $c=e$ is the threshold for the emergence of a continuous component at zero in the limiting spectral measure of sparse Erdős–Rényi graphs. The source states that this conjecture was established by Bordenave, Lelarge and Salez through a first-order analysis of the resolvent near zero.