Nodal deficiency of random spherical harmonics in presence of boundary

TL;DR

This paper analyzes the zero density and nodal length of random Gaussian Laplace eigenfunctions on the hemisphere with boundary conditions, revealing a negative bias in nodal length compared to boundary-free models.

Contribution

It provides a precise asymptotic law for zero densities and identifies a logarithmic negative bias in nodal length for boundary-adapted random spherical harmonics.

Findings

Derived asymptotic zero density laws in short and long range regimes.

Discovered a logarithmic negative bias in total nodal length due to boundary effects.

Extended understanding of nodal geometry for eigenfunctions with boundary conditions.

Abstract

We consider a random Gaussian model of Laplace eigenfunctions on the hemisphere satisfying the Dirichlet boundary conditions along the equator. For this model we find a precise asymptotic law for the corresponding zero density functions, in both short range (around the boundary) and long range (far away from the boundary) regimes. As a corollary, we were able to find a logarithmic negative bias for the total nodal length of this ensemble relatively to the rotation invariant model of random spherical harmonics. Jean Bourgain's research, and his enthusiastic approach to the nodal geometry of Laplace eigenfunctions, has made a crucial impact in the field and the current trends within. His works on the spectral correlations (Theorem 2.2 in Krishnapur, Kurlberg and Wigman (2013)) and joint with Bombieri (Bourgain and Bombieri (2015)) have opened a door for an active ongoing research on the nodal length of functions defined on surfaces of arithmetic flavour, like the torus or the square. Further, Bourgain's work on toral Laplace eigenfunctions (Bourgain (2014)), also appealing to spectral correlations, allowed for inferring deterministic results from their random Gaussian counterparts.