# Two-level correlation function of eigenvalues for large random matrices

One can define the density of eigenvalues of a $$N\times N$$ Hermitian random matrix $$H$$ as: $$$$\rho(\lambda)=\left \langle\frac{1}{N} \operatorname{Tr} \delta(\lambda-H)\right\rangle$$$$ Where $$\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: $$$$\rho^{(2)}(\lambda, \mu)=\left \langle\frac{1}{N} \operatorname{Tr} \delta(\lambda-H) \frac{1}{N} \operatorname{Tr} \delta(\mu-H)\right\rangle$$$$

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

• 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. Jun 7, 2021 at 5:32

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 $$P(\lambda_1,\lambda_2,\ldots \lambda_N)\propto e^{-\beta W},$$ $$W=\sum_{i then the two-point correlation function $$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.