Title: Nodal statistics of graph eigenfunctions
Abstract: Understanding statistical properties of zeros of Laplacian
eigenfunctions is a program which is attracting much attention from
mathematicians and physicists. We will discuss this program in the
setting of "quantum graphs", self-adjoint differential operators
acting on functions living on a metric graph.
Numerical studies of quantum graphs motivated a conjecture that the
distribution of nodal surplus (a suitably rescaled number of zeros of
the n-th eigenfunction) has a universal form: it approaches Gaussian
as the number of cycles grows. The first step towards proving this
conjecture is a result established for graphs which are composed of
cycles separated by bridges. For such graphs we use the
nodal-magnetic theorem of the speaker, Colin de Verdiere and Weyand to
prove that the distribution of the nodal surplus is binomial with
parameters p=1/2 and n equal to the number of cycles.
Based on joint work with Lior Alon and Ram Band.