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.

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.

## Date:

Wed, 10/04/2019 - 12:00 to 13:00

## Location:

Ross 70