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:
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).
— 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
then we (heuristically) have , and so we are now projecting to the region
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.
Remark 1 From a semiclassical perspective, the original spectral projection
can be expressed in phase space (using the dual frequency variable
to
) as the ellipse
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),
and similarly
(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
and thus
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.
15 comments
Comments feed for this article
24 October, 2010 at 5:32 am
Leonid Petrov
In fact, the same approach is possible for random partitions. Here one obtains difference operators instead of differential ones. E.g., this explained in this paper by G. Olshanski: arXiv:0810.3751.
25 October, 2010 at 12:25 pm
Instalaciones Deportivas
shouldn`t it be different for random partitions?
25 October, 2010 at 1:16 pm
Leonid Petrov
Well, what exactly do you mean?
I meant that for certain ensembles of random partitions the correlation kernels are spectral projections for certain difference operators. This heuristics (carried out rigorously in that paper) allows to obtain bulk and edge asymptotics.
23 November, 2010 at 10:47 pm
Anonymous
Dear Prof. Tao,
in lemma 2, how can we show that the integral exists in the definition of Airy function?
Thanks
24 November, 2010 at 9:08 am
Terence Tao
A single integration by parts will convert the improper integral into an absolutely convergent one.
24 November, 2010 at 12:04 pm
Anonymous
Do we need to use complex analysis or just
and
? I am still having problem in that case…
thank you
24 November, 2010 at 1:29 pm
Terence Tao
Use
and
. The basic point is that
is an approximate antiderivative of
, in analogy with the fact that
is an antiderivative of
.
More generally, when using integration by parts to deal with a highly oscillatory term, one wants to “integrate” the oscillatory component and differentiate the more slowly varying components in order to fully exploit cancellation. In some cases this means that one has to multiply and divide by a chain rule factor in order to easily obtain the required integration.
24 November, 2010 at 2:26 pm
Anonymous
Thanks Prof. Tao. you are great. I see it now…
26 January, 2011 at 7:01 pm
Anonymous
Dear Prof. Tao,
Here when use integration by parts, do we assume that the integration is improper Riemann integral? is integration by parts true for Lebesque integral for finite or infinite domains?
Thanks
27 January, 2011 at 1:53 am
Terence Tao
There are many rigorous justifications of the integration by parts formula under a variety of hypotheses; there is no one “best” formulation here, but rather a family of such formulations. It is not just a matter of choosing the nature of the integral and selecting the domain; some regularity and decay conditions on the functions in the formula must also be assumed (e.g. on compact smooth domains, continuous differentiability would suffice. In such cases there is no distinction between the Lebesgue and Riemann integral). Some other examples of integration by parts theorems are given in https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/
4 December, 2010 at 6:47 pm
Anonymous
Dear Prof. Tao,
In the second formula the summation is over i_1 < i_2 < .., but I guess they would appear as subscripts for the lambda's as well. Also how should one think about the ordered eigenvalue distribution? I try to form an analogy with a set of n uniform iid points on the interval. There one gets for ordered X_i's, P(X_1 = dy_1, \ldots, X_k = dy_k) = dy_1 .. dy_k, and unordered X_i's P(X_1 =dy_1, \ldots, X_k = dy_k) = n!/(n-k)! dy_1 .. dy_k, since the latter random vector lives on the simplex {x_1 < x_2 < .. < x_k}. I guess my question is does the notion of ordered probability distribution for eigenvalues have a natural interpretation?
4 December, 2010 at 7:54 pm
Terence Tao
Thanks for the correction!
One can certainly view the ordered eigenvalues as a probability distribution on the simplex
; the density function of this distribution is
. The notion of a k-point correlation function, though, is best interpreted using the unordered formalism. (In some sense, the eigenvalues behave a little bit like quantum-mechanical bosons, in that the mathematics is a bit simpler if one views them as interchangable with each other, though the analogy is not perfect.)
19 December, 2010 at 1:14 am
Samuel Monnier
I think there is a typo in the Gaudin-Mehta formula (1). The indices i and j on the right-hand side should be bounded above by k instead of n.
[Corrected, thanks – T.]
24 December, 2010 at 12:31 am
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