  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.
