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

Nodal deficiency of random spherical harmonics in presence of boundary