CLT for small scale mass distribution of toral Laplace eigenfunctions

In this talk we discuss the fine scale $L^2$-mass distribution of toral Laplace
eigenfunctions with respect to random position. For the 2-dimensional torus, under
certain flatness assumptions on the Fourier coefficients of the eigenfunctions and
generic restrictions on energy levels, both the asymptotic shape of the variance
and the limiting Gaussian law are established, in the optimal Planck-scale regime.
We also discuss the 3-dimensional case, where the asymptotic behaviour of the variance
is analysed in a more restrictive scenario. This is joint work with Igor Wigman.