Let 
be a large integer, and let 
be the Gaussian Unitary Ensemble (GUE), i.e. the random Hermitian matrix with probability distribution
where 
is a Haar measure on Hermitian matrices and 
is the normalisation constant required to make the distribution of unit mass. The eigenvalues 
of this matrix are then a coupled family of real random variables. For any 
, we can define the 
-point correlation function 
to be the unique symmetric measure on 
such that
A standard computation (given for instance in these lecture notes of mine) gives the Ginebre formula
for the -point correlation function, where 
is another normalisation constant. Using Vandermonde determinants, one can rewrite this expression in determinantal form as
where the kernel 
is given by
where 
and 
are the (
-normalised) Hermite polynomials (thus the 
are an orthonormal family, with each 
being a polynomial of degree ). Integrating out one or more of the variables, one is led to the Gaudin-Mehta formula
(In particular, the normalisation constant 
in the previous formula turns out to simply be equal to 
.) Again, see these lecture notes for details.
The functions 
can be viewed as an orthonormal basis of eigenfunctions for the harmonic oscillator operator
indeed it is a classical fact that
As such, the kernel can be viewed as the integral kernel of the spectral projection operator ![{1_{(-\infty,n+\frac{1}{2}]}(L)}](https://s0.wp.com/latex.php?latex=%7B1_%7B%28-%5Cinfty%2Cn%2B%5Cfrac%7B1%7D%7B2%7D%5D%7D%28L%29%7D&bg=ffffff&fg=000000&s=0 "{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 as 
. The two main asymptotics of interest are given by the following lemmas:
Lemma 1 (Asymptotics of in the bulk) Let 
, and let 
be the semicircular law density at 
. Then, we have
as for any fixed 
(removing the singularity at 
in the usual manner).
Lemma 2 (Asymptotics of at the edge) We have
as for any fixed , where 
is the Airy function
and again removing the singularity at 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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