Sine kernel universality, page 8

Here is an actual statement of sine kernel universality: (Probably some additional hypotheses are needed)

Choose any unitary matrix ensemble, and let \psi be its equilibrium measure. If \lambda is an interior point of the support of \psi , then we get convergence of the folowing scaled two point correlation function

\rho^{(2)}_N\left(\lambda+\frac{x}{N},\lambda+\frac{y}{N}\right) \rightarrow \psi(\lambda)^2 \left(1 - \frac{sin[\pi\psi(\lambda)(x-y)]^2}{[\pi\psi(\lambda)(x-y)]^2}\right)

pointwise as N \rightarrow \infty.

This animation shows the corresponding statement for scaled conditional eigenvalue distributions.