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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).

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