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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

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    $\begingroup$ This is a very wide question, essentially asking "what has happened in RMT since 1995?". The answer is a l"a lot", for example, universality. You should be more specific (as in the paper you quote) of the regime you care about: are $\lambda$ and $\mu$ separated (in which case you need to rescale to get interesting answers) or at distance $1/N$ of each other? For the latter case, look up work of Erdos-Yau and co-authors, as well as Tao-Vu. $\endgroup$ Jun 7 '21 at 5:32
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Let me address the issue raised by the OP of the universality of the two-point correlation function.

The universality of $\rho^{(2)}(\lambda,\mu)$ does exist if one considers the correlations locally, on the scale of the mean eigenvalue spacing. This is relevant for many applications, because the correlations decay quickly with increasing $|\lambda-\mu|$.

If the eigenvalue distribution has the Gibbs form $$P(\lambda_1,\lambda_2,\ldots \lambda_N)\propto e^{-\beta W},$$ $$W=\sum_{i<j}u(\lambda_i-\lambda_j)+\sum_i V(\lambda_i),$$ then the two-point correlation function $$K(\lambda,\mu)=N^2\rho^{(2)}(\lambda,\mu)-N^2\rho(\lambda)\rho(\mu)$$ is given in the large-$N$ local limit by the functional inverse $u^{\text{inv}}$ of $u$, $$K(\lambda,\mu)=\frac{1}{\beta}u^{\text{inv}}(\lambda,\mu).$$ See section 1.D of arXiv:cond-mat/9612179 .

Notice that this is independent of the "potential" $V$, only the eigenvalue interaction $u$ enters. That is the sense in which the two-point correlation function is "universal" on the local scale for a broad class of RMT ensembles with a logarithmic eigenvalue repulsion, $u(\lambda,\mu)=-\log|\lambda-\mu|$, see for example arXiv:cond-mat/9310010.

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