  We investigate to what extent the microscopic Wigner-Gaudin-Mehta-Dyson
(WGMD) (or sine kernel) statistics of random matrix theory remain valid on
mesoscopic scales. To that end, we compute the connected two-point spectral
correlation function of a Wigner matrix at two mesoscopically separated points.
In the mesoscopic regime, density correlations are much weaker than in the
microscopic regime. Our result is an explicit formula for the two-point
function. This formula implies that the WGMD statistics are valid to leading
order on all mesoscopic scales, that in the real symmetric case there are
subleading corrections matching precisely the WGMD statistics, while in the
complex Hermitian case these subleading corrections are absent. We also uncover
non-universal subleading correlations, which dominate over the universal ones
beyond a certain intermediate mesoscopic scale. The proof is based on a
hierarchy of Schwinger-Dyson equations for a sufficiently large class of
polynomials in the entries of the Green function. The hierarchy is indexed by a
tree, whose depth is controlled using stopping rules. A key ingredient in the
derivation of the stopping rules is a new estimate on the density of states,
which we prove to have bounded derivatives of all order on all mesoscopic
scales.
