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.
The basic tool here is cofactor expansion, which we write as follows. Given an matrix , we can expand the determinant as
where is the bottom left row of , is the top left minor of , and is the matrix formed by replacing the row of by .
As an example of how this cofactor expansion is used in determinantal point processes, let us recall Gaudin’s lemma, which we state slightly informally:
For each , let be the correlation function
Then we have
Proof: From (4) we have
where and . Integrating using (6), we conclude that
By the multilinearity of determinant, we can write
But from (5) we have
and the claim follows.
The sine kernel actually does obey (5), but we will need the following more general identity:
(using the principal value interpretation of the integral) when are distinct. In particular, by l’Hopital’s rule one has
Proof: This can be verified by a routine contour integration, but we will give a more Fourier analytic proof instead. We will omit some (routine) details concerning justifying the various interchanges of integrals we will be using here. By a limiting argument we may take to be distinct. Observe that
and so the left-hand side of (9) can be rewritten as
From the basic identity we have
for any , so the preceding expression simplifies to
For (12), we make the change of variables and rewrite this as
We have (in the distributional sense at least) that
and so by (9) this term simplifies to .
Now we turn to (13). Here we observe that in the region of integration we have , and so this expression can be factored as
The final factor here is thanks to (9), while a brief computation shows (in the distributional sense, at least) that
and the claim follows.
Now we turn to the verification of (3). It will suffice to establish the identity
for each . By a continuity argument we may take the to be distinct.
when , and
when . In the former case, we see from row operations that
while from the Leibniz rule and symmetry (and (8)) we have
putting all of these facts together, we obtain (14) as required.
Remark 1 Presumably, a similar exercise can be performed for the Airy process that governs the asymptotics of extreme eigenvalues of GUE (which include the famous Tracy-Widom law), but I did not attempt this.