One can define the density of eigenvalues of a $N\times N$ Hermitian random matrix $H$ as: \begin{equation} \rho(\lambda)=\left \langle\frac{1}{N} \operatorname{Tr} \delta(\lambda-H)\right\rangle \end{equation} Where $\langle \dots \rangle$ denotes the average over the distribution of $H$.

In the large $N$ limit, it is famously known that $\rho(\lambda)$ will approach the Wigner semi circular law (given some conditions on the moments of the distribution of $H$). This can be shown with various methods, one that I favor is using the resolvent and computing its schur complements.

One can define the two-level correlation function: \begin{equation} \rho^{(2)}(\lambda, \mu)=\left \langle\frac{1}{N} \operatorname{Tr} \delta(\lambda-H) \frac{1}{N} \operatorname{Tr} \delta(\mu-H)\right\rangle \end{equation}

The following paper 1 provides a method to compute this quantity, and provides an exhaustive list of the existing methods to compute eq $(2)$. However the paper is now $25$ years old: is there any known results that extends their method? **Can we compute this quantity using the schur complement and the resolvent?** What about non-hermitian matrices?

**Edit**:
My initial query was whether we could could compute equation $(2)$ using the resolvents and the Schur complements.

The paper mentions that contrary to $\rho(\lambda)$ there is no universality for $\rho^{(2)}(\lambda, \mu)$: it would depend on the choice of distribution. However in a certain regime, with large $N$ and small $\lambda- \mu$ then universal properties can be derived. Has this been we shown this using the Schur complements of the resolvents?

1 : Brézin, E., & Hikami, S. (1996). Correlations of nearby levels induced by a random potential. Nuclear Physics B, 479(3), 697-706. link: https://arxiv.org/pdf/cond-mat/9605046.pdf