Universality of nodal count distribution in large metric graphs

TL;DR

This paper investigates the distribution of nodal surplus in eigenfunctions of Laplacians on large metric graphs, conjecturing and providing evidence that it converges to a Gaussian distribution as the graph complexity increases.

Contribution

It introduces a conjecture that the nodal surplus distribution becomes Gaussian for large graphs and proves this for specific graph sequences, supported by numerical tests.

Findings

Distribution converges to Gaussian for certain graph sequences

Derived a formula for computing the distribution as an integral over a torus

Numerical evidence supports the conjecture for various graph families

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 $\beta$. 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 $\beta$. 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.