The Wigner-Dyson-Mehta Bulk Universality Conjecture for Wigner Matrices

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. As an application, we determine the limiting distribution of the number of eigenvalues $N_I$ in a short interval $I$ of length $\Theta (1/n)$. As a corollary of this result, we obtain an extension of a result of Jimbo et. al. concerning the behavior of spacing in the bulk.