Nodal deficiency of random spherical harmonics in presence of boundary

Valentina Cammarota,Domenico Marinucci,Igor Wigman

doi:10.1063/5.0036084

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 relative 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 the work of Krishnapur et al. [Ann. Math. 177(2), 699–737 (2013)]} and Bombieri and Bourgain [Int. Math. Res. Not. (IMRN) 11, 3343–3407 (2015)] have opened a door for an active ongoing research on the nodal length of functions defined on surfaces of arithmetic flavor, such as the torus or the square. Furthermore, Bourgain’s work [J. Bourgain, Isr. J. Math. 201(2), 611–630 (2014)] on toral Laplace eigenfunctions, also appealing to spectral correlations, allowed for inferring deterministic results from their random Gaussian counterparts.

Highlights

The Laplace eigenfunctions with eigenvalue 4π2n all correspond to an integer n expressible as a sum of two squares, and are given by a sum fn(x) =
It makes sense to compare the torus to the square with Dirichlet boundary, and test what kind of impact it would have relatively to (2.2) on the expected nodal length, as the “boundary-adapted arithmetic random waves”, that were addressed in [11]
We would like to apply (6.1) to the boundary-adapted random spherical harmonics Theorem for L (Tl) to evaluate the asymptotic law of the total expected nodal length of Tl

Summary

Introduction

An instrumental key input to both the said asymptotic variance and the limit law was Bourgain’s first nontrivial upper bound [20, Theorem 2.2] of or2(n)→∞ (r2(n)4) for the number of length-6 spectral correlations, i.e. 6-tuples of lattice points {μ : μ 2 = n} summing up to 0. It makes sense to compare the torus to the square with Dirichlet boundary, and test what kind of impact it would have relatively to (2.2) on the expected nodal length, as the “boundary-adapted arithmetic random waves”, that were addressed in [11].

Results

Conclusion