Towards universality of the nodal statistics on metric graphs
The study of nodal sets of Laplace eigenfunctions has intrigued many mathematicians over the years. The nodal count problem has its origins in the works of Strum (1936) and Courant (1923) which led to questions that remained open to this day. One such question was the universal behavior of the nodal statistics. In 2002 Blum, Gnutzmann and Smilansky observed numerically that the statistical behaviors of the (properly rescaled) nodal count on planar billiards, exhibit two types of behaviors. These behaviors seemed to be independent of the specific shape of the billiards, hence universal, and were determined by the system's classification to chaotic or integrable. Proving this universality is still open.
We study an analog problem of nodal statistics on metric graphs. In this talk, I will present a conjecture of a universal nodal statistics behavior for metric graphs and will provide new experimental and analytic results supporting the conjecture. I will describe our methods of proof for certain families of graphs, exploiting the symmetries of their secular manifolds.
This is a joint work with R. Band (Technion) and G. Berkolaiko (Texas A&M).