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Let be a large natural number, and let be a matrix drawn from the Gaussian Unitary Ensemble (GUE), by which we mean that is a Hermitian matrix whose upper triangular entries are iid complex gaussians with mean zero and variance one, and whose diagonal entries are iid real gaussians with mean zero and variance one (and independent of the upper triangular entries). The eigenvalues are then real and almost surely distinct, and can be viewed as a random point process on the real line. One can then form the -point correlation functions for every , which can be defined by duality by requiring
for any test function . For GUE, which is a continuous matrix ensemble, one can also define for distinct as the unique quantity such that the probability that there is an eigenvalue in each of the intervals is in the limit .
As is well known, the GUE process is a determinantal point process, which means that -point correlation functions can be explicitly computed as
for some kernel ; explicitly, one has
Using the asymptotics of Hermite polynomials (which then give asymptotics for the kernel ), one can take a limit of a (suitably rescaled) sequence of GUE processes to obtain the Dyson sine process, which is a determinantal point process on the real line with correlation functions
A bit more precisely, for any fixed bulk energy , the renormalised point processes converge in distribution in the vague topology to as , where is the semi-circular law density.
On the other hand, an important feature of the GUE process is its stationarity (modulo rescaling) under Dyson Brownian motion
which describes the stochastic evolution of eigenvalues of a Hermitian matrix under independent Brownian motion of its entries, and is discussed in this previous blog post. To cut a long story short, this stationarity tells us that the self-similar -point correlation function
obeys the Dyson heat equation
(see Exercise 11 of the previously mentioned blog post). Note that vanishes to second order whenever two of the coincide, so there is no singularity on the right-hand side. Setting and using self-similarity, we can rewrite this equation in time-independent form as
One can then integrate out all but of these variables (after carefully justifying convergence) to obtain a system of equations for the -point correlation functions :
where the integral is interpreted in the principal value case. This system is an example of a BBGKY hierarchy.
If one carefully rescales and takes limits (say at the energy level , for simplicity), the left-hand side turns out to rescale to be a lower order term, and one ends up with a hierarchy for the Dyson sine process:
Informally, these equations show that the Dyson sine process is stationary with respect to the infinite Dyson Brownian motion
where are independent Brownian increments, and the sum is interpreted in a suitable principal value sense.
I recently set myself the exercise of deriving the identity (3) directly from the definition (1) of the Dyson sine process, without reference to GUE. This turns out to not be too difficult when done the right way (namely, by modifying the proof of Gaudin’s lemma), although it did take me an entire day of work before I realised this, and I could not find it in the literature (though I suspect that many people in the field have privately performed this exercise in the past). In any case, I am recording the computation here, largely because I really don’t want to have to do it again, but perhaps it will also be of interest to some readers.
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).