DysonMehtaTheorem

Correlation functions

The joint probability density of the eigenvalues of $n\times n$ random Hermitian matrices $M \in \mathbf{H}^{n \times n}$, with partition functions of the form

$$Z_n = \int_{M \in \mathbf{H}^{n \times n}}e^{\text{tr}(V(M)) d\mu_0(M)}$$

where

$$V(x):=\sum_{j=1}^\infty v_j x^j$$

and $d\mu_0(M)$ is the standard Lebesgue measure on the space $\mathbf{H}^{n \times n}$ of Hermitian $n \times n$ matrices, is given by

$$p_{n,V}(x_1, \ldots, x_n) = \left(\frac{1}{Z_{n,V}}\right) \left( \prod_{i=1}^{n-1} \prod_{j=i+1}^{n} (x_i - x_j)^2 \right) \exp\left(-\sum_{i=1}^{n} V(x_i)\right)$$

The $k$-point correlation functions (or marginal distributions) are defined as

$$R^{(k)}_{n,V}(x_1,\dots,x_k) = \frac{n!}{(n-k)!} \int_{\mathbf{R}}dx_{k+1} \cdots \int_{\mathbf{R}} dx_{n} p_{n,V}(x_1,x_2,\dots,x_n),$$

which are skew symmetric functions of their variables. In particular, the one-point correlation function, or density of states, is

$$R^{(1)}_{n,V}(x_1) = n\int_{\mathbf{R}}dx_{2} \cdots \int_{\mathbf{R}} dx_{n} p_{n,V}(x_1,x_2,\dots,x_n).$$

Its integral over a Borel set $B \subset \mathbf{R}$ gives the expected number of eigenvalues contained in $B$:

$$\int_{B} R^{(1)}_{n,V}(x)dx = \mathbf{E}\left(\#\{\text{eigenvalues in }B\}\right).$$

The following result expresses these correlation functions as determinants of the matrices formed from evaluating the appropriate integral kernel at the pairs $(x_i, x_j)$ of points appearing within the correlator.

Theorem [Dyson-Mehta] For any $k$, $1\leq k \leq n$ the $k$-point correlation function $R^{(k)}_{n,V}$ can be written as a determinant

$$R^{(k)}_{n,V}(x_1,x_2,\dots,x_k) = \det_{1\leq i,j \leq k}\left(K_{n,V}(x_i,x_j)\right),$$

where $K_{n,V}(x,y)$ is the $n$-th Christoffel-Darboux kernel

$$K_{n,V}(x,y) := \sum_{k=0}^{n-1}\psi_k(x)\psi_k(y),$$

associated to $V$, written in terms of the quasipolynomials

$$\psi_k(x) = {1\over \sqrt{h_k}}\, p_k(z)\, e^{- V(z) / 2}$$

where ${p_k(x)}_{k\in \mathbf{N}}$ is a complete sequence of monic polynomials, of the degrees indicated, satisfying the orthogonilty conditions

$$\int_{\mathbf{R}} \psi_j(x) \psi_k(x) dx = \delta_{jk}$$