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. Contents1. Introduction 2. Definitions and preliminaries 3. Conjecture and main results 3.1. Analytical evidence in support of Conjecture 3.1 3.2. Efficient computation of the nodal surplus distribution 3.3. The polytope of nodal surplus distributions of a given graph 3.4. Numerical evidence in support of Conjecture 3.1 4. Secular averages 4.1. The secular manifold 4.2. The Barra-Gaspard method 4.3. BG method as a torus intergal 5. The nodal surplus distribution 5.1. The nodal surplus oracle function 5.2. Hessian in terms of the unitary evolution matrices 5.3. Proof of Theorem 3.7 and its generalization to graphs with loops. 6. The polytope of nodal surplus distributions and discrete graphs symmetries. 7. Mandarins and stowers -proof of Theorem 3.4