Universality conjecture for nodal surplus distributions on large metric graphs

Let $\left\{\Gamma^{(\beta)}\right\}_{\beta\nearrow\infty}$ be any sequence of standard graphs, labeled by their first Betti numbers. For each $\Gamma^{(\beta)}$, choose arbitrary rationally independent edge lengths, and let $\sigma^{(\beta)}$ denote its nodal surplus random variable. Nodal surplus universality conjecture. As $\beta\to\infty$, the normalized surplus satisfies

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

where $N(0,1)$ is the standard normal distribution. Moreover, there is a constant $C>1$ such that, for all sufficiently large $\beta$,

$$\frac{\beta}{C}\leq\operatorname{Var}(\sigma^{(\beta)})\leq C\beta.$$

This conjecture asserts a universal Gaussian limit for nodal surplus statistics of large standard metric graphs, independently of their topology and rationally independent edge lengths. The preceding results establish that the surplus distribution is well defined and symmetric with mean $\beta/2$; the Gaussian convergence and linear variance bounds are the proposed claims.