Universality of Nodal Count Distribution in Large Metric Graphs

Abstract

An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph’s non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph’s first Betti number β. We study the distribution of the nodal surplus values in the countably infinite set of the graph’s eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing β. We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computation of the distribution is made possible by a formula expressing the nodal surplus distribution as an integral over a high-dimensional torus.

KEYWORDS:

• Quantum graphs
• nodal count
• quantum chaos

Acknowledgments

We thank the referees for valuable comments. We thank Ronen Eldan for interesting discussions.

Notes

1 Only relatively recently it was extended to Neumann Laplacians [42, 50].

2 In the non-generic case, the behavior of ${\nu }_n/n$ becomes highly non-trivial. Recent progress was made in [35], showing that any sub-sequential limit of ${\nu }_n/n$ is given as a ratio between the length of a sub-graph and the length of the entire graph. This provides lower and upper bounds on the possible limits.

3 Also called natural or standard vertex conditions.

4 The large graphs limit in [33] is $V\to \infty$ for “well connected” graphs.

5 also known as pumpkin or watermelon.

6 We apply Bernstein inequality for bounded zero-mean independent random variables. We consider the random variables $X_i:=f({\overrightarrow{\kappa }}_i)-P({\omega }_e=s)$, which have zero-mean and are bounded by $\left|X_i\right|\le 1$.

7 Even real analytic [4].

8 An oracle depending only on $\overrightarrow{\kappa }$ has no access to the label n of the eigenfunction which enters the definition of the nodal surplus. The label is highly sensitive to the changes in the edge lengths $\mathbf{\ell }$. Remarkably, taking the difference in (2.1) erases this dependence on $\mathbf{\ell }$.

9 ${\Sigma }^{\text{reg}}$ is an orientable Riemannian manifold (with metric inherited by the flat metric on ${\mathbb{T}}^E$) and as such has a standard volume form.

10 The normal can be chosen to have non-negative entries by [25, Theorem 1.1].

11 By Riemann integrable we mean that its discontinuity set has measure zero.

12 The definitions of $\mathit{\sigma }$ in [5] and in the present paper are slightly different but agree when restricted to ${\Sigma }^{\mathcal{G}}$

13 That is, any pair of edges $e,e\prime$ has a graph automorphism $g\in G_{\Gamma }$ such that $g(e)=e\prime$.

14 By pull-back of the uniform measure we mean that their push-forward is the uniform (Lebesgue) measure.

15 Using $|a-b|<c\Rightarrow -c<\left|a\right|-\left|b\right|<c$ (the “dark side” of the triangle inequality).

16 Given a matrix A with eigenvalue λ of algebraic multiplicity m. For small enough $ϵ>0$ there is a δ such that any $A\prime$ in a δ neighborhood of A has exactly m eigenvalues (counting with algebraic multiplicity) in an $ϵ$ ball around λ. To see that, apply Rouché’s theorem to characteristic polynomials.

17 we may assume that M is a connected finite dimensional manifold and hence path connected.

18 For this to make sense we should take the neighborhoods small enough such that each $\left\{t\in [0,1]:\gamma (t)\in {\Omega }_j\right\}$ is a connected open interval, and none of these intervals is completely contained in another.

Additional information

Funding

The authors were supported by the Binational Science Foundation Grant (Grant No. 2016281). RB and LA were also supported by ISF (Grant No. 844/19). LA was also supported by the Ambrose Monell Foundation and the Institute for Advanced Study. GB was also supported by the National Science Foundation (DMS-1815075).