Al-Obeid–Band–Berkolaiko nodal-surplus Gaussian-limit conjecture

Let

$$be the first Betti number of a quantum graph with rationally independent edge lengths, and let$$

be its nodal surplus random variable. Write $\operatorname{Var}(\sigma)$ for its variance. Al-Obeid–Band–Berkolaiko's conjecture. As $\to\infty$,

$$\frac{\sigma-\frac{\beta}{2}}{\sqrt{\operatorname{Var}(\sigma)}}\xrightarrow[\beta\to\infty]{\mathcal{D}}N(0,1),$$

where the convergence is in distribution and

$$\operatorname{Var}(\sigma)=\mathcal{O}(\beta).$$

The conjecture modifies the earlier nodal-statistics conjecture by using the nodal surplus and the first Betti number. The source proves the Gaussian limit for trees of cycles, but presents the general formulation as a conjecture.