  We study the fine scale $L^2$-mass distribution of toral Laplace
eigenfunctions with respect to random position, in 2 and 3 dimensions. In 2d,
under certain flatness assumptions on the Fourier coefficients and generic
restrictions on energy levels, both the asymptotic shape of the variance is
determined and the limiting Gaussian law is established, in the optimal
Planck-scale regime. In 3d the asymptotic behaviour of the variance is analysed
in a more restrictive scenario ("Bourgain's eigenfunctions"). Other than the
said precise results, lower and upper bounds are proved for the variance, under
more general flatness assumptions on the Fourier coefficients.
