  A well known conjecture of Wigner, Dyson, and Mehta asserts that the
(appropriately normalized) $k$-point correlation functions of the eigenvalues
of random $n \times n$ Wigner matrices in the bulk of the spectrum converge (in
various senses) to the $k$-point correlation function of the Dyson sine process
in the asymptotic limit $n \to \infty$. There has been much recent progress on
this conjecture, in particular it has been established under a wide variety of
decay, regularity, and moment hypotheses on the underlying atom distribution of
the Wigner ensemble, and using various notions of convergence. Building upon
these previous results, we establish new instances of this conjecture with
weaker hypotheses on the atom distribution and stronger notions of convergence.
In particular, assuming only a finite moment condition on the atom
distribution, we can obtain convergence in the vague sense, and assuming an
additional regularity condition, we can upgrade this convergence to locally
$L^1$ convergence.
