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Nodal deficiency of random spherical harmonics in presence of boundary

Research output: Contribution to journalArticlepeer-review

Original languageEnglish
Accepted/In press27 Dec 2020

King's Authors


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 [M. Krishnapur-P. Kurlberg-I. Wigman, Ann. Math. 2013]) and [E. Bombieri-J. Bourgain, IMRN 2014] 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 [J. Bourgain, Isr. J. Math. 2014] on toral Laplace eigenfunctions, also appealing to spectral correlations, allowed for inferring deterministic results from their random Gaussian counterparts.

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