Wigner–Dyson–Mehta bulk universality conjecture for Wigner matrices

Let $M_n$ be a Wigner Hermitian matrix, fix $k\geq 1$, a bulk energy $-2<u<2$, and atom distributions $\xi,\tilde\xi$. Write $\rho_{n,u}^{(k)}$ for the normalized $k$-point correlation function at energy $u$. Define the Dyson sine kernel by

$$K_{\operatorname{Sine}}(t,t'):=\frac{\sin(\pi(t'-t))}{\pi(t'-t)}.$$

The associated Dyson sine kernel $k$-point correlation functions are

$$\rho_{\operatorname{Sine}}^{(k)}(t_1,\ldots,t_k):=\det\bigl(K_{\operatorname{Sine}}(t_i,t_j)\bigr)_{1\leq i,j\leq k}.$$

Wigner–Dyson–Mehta bulk universality conjecture. For fixed $k$, $u$, and atom distributions, $\rho_{n,u}^{(k)}$ converges to $\rho_{\operatorname{Sine}}^{(k)}$ as $n{\tilde o}\infty$.

This conjecture asserts that local eigenvalue statistics in the bulk are universal, independent of the atom distributions of the Wigner ensemble. It has been established under a wide variety of decay, regularity, and moment hypotheses, and using various notions of convergence; the statement as formulated for arbitrary atom distributions is therefore recorded as solved.