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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 .
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:
as for any fixed (removing the singularity at in the usual manner).
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).