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).

— 1. The bulk asymptotics —

We begin with the bulk asymptotics, Lemma 1. Fix {x_0} in the bulk region {(-2,2)}. Applying the change of variables

\displaystyle  x = x_0 \sqrt{n} + \frac{y}{\sqrt{n} \rho_{sc}(x_0)}

we see that the harmonic oscillator {L} becomes

\displaystyle  - n \rho_{sc}(x_0)^2 \frac{d^2}{dy^2} + \frac{1}{4} (x_0 \sqrt{n} + \frac{y}{\sqrt{n} \rho_{sc}(x_0)})^2

Since {K_n} is the integral kernel of the spectral projection to the region {L \leq n+\frac{1}{2}}, we conclude that the left-hand side of (2) (as a function of {y,z}) is the integral kernel of the spectral projection to the region

\displaystyle  - n \rho_{sc}(x_0)^2 \frac{d^2}{dy^2} + \frac{1}{4} (x_0 \sqrt{n} + \frac{y}{\sqrt{n} \rho_{sc}(x_0)})^2 \leq n + \frac{1}{2}.

Isolating out the top order terms in {n}, we can rearrange this as

\displaystyle  -\frac{d^2}{dy^2} \leq \pi^2 + o(1).

Thus, in the limit {n \rightarrow \infty}, we expect (heuristically, at least) that the left-hand side of (2) to converge as {n \rightarrow \infty} to the integral kernel of the spectral projection to the region

\displaystyle  -\frac{d^2}{dy^2} \leq \pi^2.

Introducing the Fourier dual variable {\xi} to {y}, as manifested by the Fourier transform

\displaystyle  \hat f(\xi) = \int_{\bf R} e^{-2\pi i \xi y} f(y)\ dy

and its inverse

\displaystyle  \check F(y) = \int_{\bf R} e^{2\pi i \xi y} F(\xi)\ d\xi,

then we (heuristically) have {\frac{d}{dy} = 2\pi i \xi}, and so we are now projecting to the region

\displaystyle  |\xi|^2 \leq 1/4, \ \ \ \ \ (4)

i.e. we are restricting the Fourier variable to the interval {[-1/2,1/2]}. Back in physical space, the associated projection {P} thus takes the form

\displaystyle  P f(y) = \int_{[-1/2,1/2]} e^{2\pi i \xi y} \hat f(\xi)\ d\xi

\displaystyle  = \int_{\bf R} \int_{[-1/2,1/2]} e^{2\pi i \xi y} e^{-2\pi i \xi z}\ d\xi f(z)\ dz

\displaystyle  = \int_{\bf R} \frac{\sin(\pi(y-z))}{y-z} f(z)\ dz

and the claim follows.

Remark 1 From a semiclassical perspective, the original spectral projection {L \leq n+\frac{1}{2}} can be expressed in phase space (using the dual frequency variable {\eta} to {x}) as the ellipse

\displaystyle  4 \pi^2 \eta^2 + \frac{x^2}{4} \leq n+\frac{1}{2} \ \ \ \ \ (5)

which after the indicated change of variables becomes the elongated ellipse

\displaystyle  \xi^2 + \frac{1}{2n \rho_{sc}(x_0)(4-x_0^2)} y + \frac{1}{4n^2 \rho_{sc}(x_0)^2 (4-x_0^2)} y^2

\displaystyle \leq \frac{1}{4} + \frac{1}{2n (4-x_0^2)}

which converges (in some suitably weak sense) to the strip (4) as {n \rightarrow \infty}.

— 2. The edge asymptotics —

A similar (heuristic) argument gives the edge asymptotics, Lemma 2. Starting with the change of variables

\displaystyle  x = 2 \sqrt{n} + \frac{y}{n^{1/6}}

the harmonic oscillator {L} now becomes

\displaystyle  - n^{1/3} \frac{d^2}{dy^2} + \frac{1}{4} (2 \sqrt{n} + \frac{y}{n^{1/6}})^2.

Thus, the left-hand side of (3) becomes the kernel of the spectral projection to the region

\displaystyle  - n^{1/3} \frac{d^2}{dy^2} + \frac{1}{4} (2 \sqrt{n} + \frac{y}{n^{1/6}})^2 \leq n + \frac{1}{2}.

Expanding out, computing all terms of size {n^{1/3}} or larger, and rearranging, this (heuristically) becomes

\displaystyle  - \frac{d^2}{dy^2} + y \leq o(1)

and so, heuristically at least, we expect (3) to converge to the kernel of the projection to the region

\displaystyle  - \frac{d^2}{dy^2} + y \leq 0. \ \ \ \ \ (6)

To compute this, we again pass to the Fourier variable {\xi}, converting the above to

\displaystyle  4 \pi^2 \xi^2 + \frac{1}{2\pi i} \frac{d}{d\xi} \leq 0

using the usual Fourier-analytic correspondences between multiplication and differentiation. If we then use the integrating factor transformation

\displaystyle  F(\xi) \mapsto e^{8 \pi^3 i \xi^3 / 3} F(\xi)

we can convert the above region to

\displaystyle  \frac{1}{2\pi i} \frac{d}{d\xi} \leq 0

which on undoing the Fourier transformation becomes

\displaystyle  y \leq 0,

and the spectral projection operation for this is simply the spatial multiplier {1_{(-\infty,0]}}. Thus, informally at least, we see that the spectral projection {P} to the region (6) is given by the formula

\displaystyle  P = M^{-1} 1_{(-\infty,0]} M

where the Fourier multiplier {M} is given by the formula

\displaystyle  \widehat{Mf}(\xi) := e^{8 \pi^3 i \xi^3 / 3} \hat f(\xi).

In other words (ignoring issues about convergence of the integrals),

\displaystyle  Mf(y) = \int_{\bf R} (\int_{\bf R} e^{2\pi i y \xi} e^{8 \pi^3 i \xi^3 / 3} e^{-2\pi i z \xi}\ d\xi) f(z)\ dz

\displaystyle  = 2 \int_{\bf R} (\int_0^\infty \cos( 2\pi (y-z) \xi + 8 \pi^3 \xi^3 / 3 )\ d\xi) f(z)\ dz

\displaystyle  = \frac{1}{\pi} \int_{\bf R} (\int_0^\infty \cos( t (y-z) + t^3 / 3 )\ dt) f(z)\ dz

\displaystyle  = \int_{\bf R} Ai(y-z) f(z)\ dz

and similarly

\displaystyle  M^{-1} f(z) = \int_{\bf R} Ai(y-z) f(y)\ dy

(this reflects the unitary nature of {M}). We thus see (formally, at least) that

\displaystyle  P f(y) = \int_{\bf R} (\int_{(-\infty,0]} Ai(y-w) Ai(z-w)\ dw) f(z)\ dz.

To simplify this expression we perform some computations closely related to the ones above. From the Fourier representation

\displaystyle  Ai(y) = \frac{1}{\pi} \int_0^\infty \cos(ty + t^3/3)\ dt

\displaystyle  = \int_{\bf R} e^{2\pi i y \xi} e^{8 \pi i \xi^3/3}\ d\xi

we see that

\displaystyle  \widehat{Ai}(\xi) = e^{8 \pi^3 i \xi^3/3}

which means that

\displaystyle  (4 \pi^2 \xi^2 + \frac{1}{2\pi i} \frac{d}{d\xi}) \widehat{Ai}(\xi) = 0

and thus

\displaystyle  (- \frac{d^2}{dy^2} + y) Ai(y) = 0,

thus {Ai} obeys the Airy equation

\displaystyle  Ai''(y) = y Ai(y).

Using this, one soon computes that

\displaystyle  \frac{d}{dw} \frac{Ai(y-w) Ai'(z-w) - Ai'(y-w) Ai(z-w)}{y-z} = Ai(y-w) Ai(z-w).

Also, stationary phase asymptotics tell us that {Ai(y)} decays exponentially fast as {y \rightarrow +\infty}, and hence {Ai(y-w)} decays exponentially fast as {w \rightarrow -\infty} for fixed {y}; similarly for {Ai'(z-w), Ai'(y-w), Ai(z-w)}. From the fundamental theorem of calculus, we conclude that

\displaystyle  \int_{(-\infty,0]} Ai(y-w) Ai(z-w)\ dw = \frac{Ai(y) Ai'(z) - Ai'(y) Ai(z)}{y-z},

(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

\displaystyle  4 \pi^2 n^{1/3} \xi^2 + \frac{(2\sqrt{n} + y/n^{1/6})^2}{4} \leq n+\frac{1}{2},

which we can rearrange as the eccentric ellipse

\displaystyle  4\pi^2 \xi^2 + y \leq \frac{1}{2n^{1/3}} - \frac{y^2}{4 n^{2/3}}

which is converging as {n \rightarrow \infty} to the parabolic region

\displaystyle  4\pi^2 \xi^2 + y \leq 0

which can then be shifted to the half-plane {y \leq 0} by the parabolic shear transformation {(y,\xi) \mapsto (y+4\pi^2\xi^2,\xi)}, which is the canonical relation of the Fourier multiplier {M}. (The rapid decay of the kernel {Ai} of {M} at {+\infty} 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

\displaystyle  C_n e^{-\hbox{tr}(P(M_n))} dM_n

for some potential function {P}. 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.