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).
— 1. The bulk asymptotics —
We begin with the bulk asymptotics, Lemma 1. Fix in the bulk region . Applying the change of variables
we see that the harmonic oscillator becomes
Since is the integral kernel of the spectral projection to the region , we conclude that the left-hand side of (2) (as a function of ) is the integral kernel of the spectral projection to the region
Isolating out the top order terms in , we can rearrange this as
Thus, in the limit , we expect (heuristically, at least) that the left-hand side of (2) to converge as to the integral kernel of the spectral projection to the region
Introducing the Fourier dual variable to , as manifested by the Fourier transform
and its inverse
i.e. we are restricting the Fourier variable to the interval . Back in physical space, the associated projection thus takes the form
and the claim follows.
which after the indicated change of variables becomes the elongated ellipse
which converges (in some suitably weak sense) to the strip (4) as .
— 2. The edge asymptotics —
A similar (heuristic) argument gives the edge asymptotics, Lemma 2. Starting with the change of variables
the harmonic oscillator now becomes
Thus, the left-hand side of (3) becomes the kernel of the spectral projection to the region
Expanding out, computing all terms of size or larger, and rearranging, this (heuristically) becomes
and so, heuristically at least, we expect (3) to converge to the kernel of the projection to the region
To compute this, we again pass to the Fourier variable , converting the above to
using the usual Fourier-analytic correspondences between multiplication and differentiation. If we then use the integrating factor transformation
we can convert the above region to
which on undoing the Fourier transformation becomes
and the spectral projection operation for this is simply the spatial multiplier . Thus, informally at least, we see that the spectral projection to the region (6) is given by the formula
where the Fourier multiplier is given by the formula
In other words (ignoring issues about convergence of the integrals),
(this reflects the unitary nature of ). We thus see (formally, at least) that
To simplify this expression we perform some computations closely related to the ones above. From the Fourier representation
we see that
which means that
thus obeys the Airy equation
Using this, one soon computes that
Also, stationary phase asymptotics tell us that decays exponentially fast as , and hence decays exponentially fast as for fixed ; similarly for . From the fundamental theorem of calculus, we conclude that
(this is a continuous analogue of the Christoffel-Darboux formula), and the claim follows.
Remark 2 As in the bulk case, one can take a semi-classical analysis perspective and track what is going on in phase space. With the scaling we have selected, the ellipse (5) has become
which we can rearrange as the eccentric ellipse
which is converging as to the parabolic region
which can then be shifted to the half-plane by the parabolic shear transformation , which is the canonical relation of the Fourier multiplier . (The rapid decay of the kernel of at is then reflected in the fact that this transformation only shears to the right and not the left.)
Remark 3 Presumably one should also be able to apply the same heuristics to other invariant ensembles, such as those given by probability distributions of the form
for some potential function . Certainly one can soon get to an orthogonal polynomial formulation of the determinantal kernel for such ensembles, but I do not know if the projection operators for such kernels can be viewed as spectral projections to a phase space region as was the case for GUE. But if one could do this, this would provide a heuristic explanation as to the universality phenomenon for such ensembles, as Taylor expansion shows that all (reasonably smooth) regions of phase space converge to universal limits (such as a strip or paraboloid) after rescaling around either a non-critical point or a critical point of the region with the appropriate normalisation.