  We consider Berezin-Toeplitz operators on compact Kahler manifolds whose
symbols are characteristic functions. When the support of the characteristic
function has a smooth boundary, we prove a two-term Weyl law, the second term
being proportional to the Riemannian volume of the boundary. As a consequence,
we deduce the area law for the entanglement entropy of integer quantum Hall
states. Another application is for the determinantal processes with correlation
kernel the Bergman kernels of a positive line bundle : we prove that the number
of points in a smooth domain is asymptotically normal.
