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Let ${n}$ be a large integer, and let ${M_n}$ be the Gaussian Unitary Ensemble (GUE), i.e. the random Hermitian matrix with probability distribution

$\displaystyle C_n e^{-\hbox{tr}(M_n^2)/2} dM_n$

where ${dM_n}$ is a Haar measure on Hermitian matrices and ${C_n}$ is the normalisation constant required to make the distribution of unit mass. The eigenvalues ${\lambda_1 < \ldots < \lambda_n}$ of this matrix are then a coupled family of ${n}$ real random variables. For any ${1 \leq k \leq n}$, we can define the ${k}$-point correlation function ${\rho_k( x_1,\ldots,x_k )}$ to be the unique symmetric measure on ${{\bf R}^k}$ such that

$\displaystyle \int_{{\bf R}^k} F(x_1,\ldots,x_k) \rho_k(x_1,\ldots,x_k) = {\bf E} \sum_{1 \leq i_1 < \ldots < i_k \leq n} F( \lambda_{i_1}, \ldots, \lambda_{i_k} ).$

A standard computation (given for instance in these lecture notes of mine) gives the Ginebre formula

$\displaystyle \rho_n(x_1,\ldots,x_n) = C'_n (\prod_{1 \leq i < j \leq n} |x_i-x_j|^2) e^{-\sum_{j=1}^n |x_j|^2/2}.$

for the ${n}$-point correlation function, where ${C'_n}$ is another normalisation constant. Using Vandermonde determinants, one can rewrite this expression in determinantal form as

$\displaystyle \rho_n(x_1,\ldots,x_n) = C''_n \det(K_n(x_i,x_j))_{1 \leq i, j \leq n}$

where the kernel ${K_n}$ is given by

$\displaystyle K_n(x,y) := \sum_{k=0}^{n-1} \phi_k(x) \phi_k(y)$

where ${\phi_k(x) := P_k(x) e^{-x^2/4}}$ and ${P_0, P_1, \ldots}$ are the (${L^2}$-normalised) Hermite polynomials (thus the ${\phi_k}$ are an orthonormal family, with each ${P_k}$ being a polynomial of degree ${k}$). Integrating out one or more of the variables, one is led to the Gaudin-Mehta formula

$\displaystyle \rho_k(x_1,\ldots,x_k) = \det(K_n(x_i,x_j))_{1 \leq i, j \leq k}. \ \ \ \ \ (1)$

(In particular, the normalisation constant ${C''_n}$ in the previous formula turns out to simply be equal to ${1}$.) Again, see these lecture notes for details.

The functions ${\phi_k(x)}$ can be viewed as an orthonormal basis of eigenfunctions for the harmonic oscillator operator

$\displaystyle L \phi := (-\frac{d^2}{dx^2} + \frac{x^2}{4})\phi;$

indeed it is a classical fact that

$\displaystyle L \phi_k = (k + \frac{1}{2}) \phi_k.$

As such, the kernel ${K_n}$ can be viewed as the integral kernel of the spectral projection operator ${1_{(-\infty,n+\frac{1}{2}]}(L)}$.

From (1) we see that the fine-scale structure of the eigenvalues of GUE are controlled by the asymptotics of ${K_n}$ as ${n \rightarrow \infty}$. The two main asymptotics of interest are given by the following lemmas:

Lemma 1 (Asymptotics of ${K_n}$ in the bulk) Let ${x_0 \in (-2,2)}$, and let ${\rho_{sc}(x_0) := \frac{1}{2\pi} (4-x_0^2)^{1/2}_+}$ be the semicircular law density at ${x_0}$. Then, we have

$\displaystyle K_n( x_0 \sqrt{n} + \frac{y}{\sqrt{n} \rho_{sc}(x_0)}, x_0 \sqrt{n} + \frac{z}{\sqrt{n} \rho_{sc}(x_0)} )$

$\displaystyle \rightarrow \frac{\sin(\pi(y-z))}{\pi(y-z)} \ \ \ \ \ (2)$

as ${n \rightarrow \infty}$ for any fixed ${y,z \in {\bf R}}$ (removing the singularity at ${y=z}$ in the usual manner).

Lemma 2 (Asymptotics of ${K_n}$ at the edge) We have

$\displaystyle K_n( 2\sqrt{n} + \frac{y}{n^{1/6}}, 2\sqrt{n} + \frac{z}{n^{1/6}} )$

$\displaystyle \rightarrow \frac{Ai(y) Ai'(z) - Ai'(y) Ai(z)}{y-z} \ \ \ \ \ (3)$

as ${n \rightarrow \infty}$ for any fixed ${y,z \in {\bf R}}$, where ${Ai}$ is the Airy function

$\displaystyle Ai(x) := \frac{1}{\pi} \int_0^\infty \cos( \frac{t^3}{3} + tx )\ dt$

and again removing the singularity at ${y=z}$ in the usual manner.

The proof of these asymptotics usually proceeds via computing the asymptotics of Hermite polynomials, together with the Christoffel-Darboux formula; this is for instance the approach taken in the previous notes. However, there is a slightly different approach that is closer in spirit to the methods of semi-classical analysis, which was briefly mentioned in the previous notes but not elaborated upon. For sake of completeness, I am recording some notes on this approach here, although to focus on the main ideas I will not be completely rigorous in the derivation (ignoring issues such as convegence of integrals or of operators, or (removable) singularities in kernels caused by zeroes in the denominator).