Math 289A: Lecture 14

In this lecture I wanted to connect our (admittedly meandering) discussion so far to the standard approach to limit theorems in random matrix theory. Let X_N be a random Hermitian matrix with characteristic function

\varphi_{X_N}(T) = e^{-\frac{1}{2} \mathrm{Tr}\, T^2}, \quad T \in \mathbb{H}_N.

This means precisely that the distribution of X_N in \mathbb{H}_N with Hilbert-Schmidt scalar product is the standard Gaussian measure on this Euclidean space.

Definition 14.1. The Gaussian Unitary Ensemble is the sequence (X_N)_{N=1}^\infty.

The word “unitary” here means that the distribution of X_N is unitarily invariant. Let $B_{N1} \geq \dots \geq B_{NN}$ be the eigenvalues of X_N.

Theorem 14.1 (Wigner). For each d \in \mathbb{N}, the limit

\gamma_d = \lim\limits_{N \to \infty} \mathbf{E}\frac{p_d(\left( \frac{B_{N1}}{\sqrt{N}},\dots,\frac{B_{NN}}{\sqrt{N}}\right)}{N}

exits and is given by

\gamma_d = \begin{cases} 0, \text{ if }d \text{ odd}\\ \mathrm{Cat}(\frac{d}{2}), \text{ if }d \text{ even} \end{cases}.

In Theorem 14.1, p_d(x_1,\dots,x_N)=x_1^d+\dots+x_N^d is the power sum symmetric polynomial of degree d, and \mathrm{Cat}(k)=\frac{1}{k+1}{2k \choose k} is the kth Catalan number. There are two things we should discuss: what the result means, and how to prove it.

Concerning the meaning of Theorem 14.1, the random variable

\frac{p_d(B_{N1},\dots,B_{NN})}{N} = \frac{B_{N1}^d + \dots + B_{NN}^d}{N}

is the degree d moment of the empirical eigenvalue distribution of X_N, which is the random discrete probability measure \eta_N on \mathbb{R} which places equal mass at each eigenvalue. Thus, Theorem 14.1 is saying that each moment of the empirical eigenvalue distribution of the scaled random matrix \frac{1}{\sqrt{N}}X_N converges in expectation to an explicit limit. It is therefore natural to ask whether \gamma_1,\gamma_2,\gamma_3,\dots is the moment sequence of a probability measure on \mathbb{R}. To answer this, one can recognize that the exponential generating function

E(z) = \sum\limits_{d=0}^\infty z^d \gamma_d = \sum_{k=0}^\infty \frac{z^{2k}}{(k+1)!k!}

is essentially a Bessel function. Then, a classical integral representation for Bessel functions allows one to determine that (\gamma_d)_{d=1}^\infty is the moment sequence of the probability measure \omega on \mathbb{R} which is supported on [-2,2] with density

\omega(\mathrm{d}x) = \frac{1}{2\pi}\sqrt{4-x^2}.

You can find the details of this derivation here, towards the end of Lecture One.

Now we discuss how one can prove Theorem 14.1. The proof uses the fact that power sums in the eigenvalues of a matrix are also traces of powers of that matrix. This means that we have

\frac{p_d(\left( \frac{B_{N1}}{\sqrt{N}},\dots,\frac{B_{NN}}{\sqrt{N}}\right)}{N}=\frac{1}{N} \mathrm{Tr} (\frac{1}{\sqrt{N}}X_N)^d,

the point being that the right hand side is also a polynomial in the elements X_N(i,j) of the Gaussian random matrix X_N. Indeed, by matrix algebra we have

\mathrm{Tr}\, X_N^d = \sum\limits_\phi X_N(\phi(1),\phi(2))X_N(\phi(2),\phi(3)) \dots X_N(\phi(d),\phi(1)),

where the sum is over all functions

\phi \colon \{1,\dots,d\} \longrightarrow \{1,\dots,N\}.

Using the fact that the matrix elements of X_N are Gaussian and independent (up to selfadjointness), the following can be established.

Theorem 14.2 (Harer-Zagier). For each d \in \mathbb{N}, we have

\mathbf{E}\left[\frac{1}{N} \mathrm{Tr} (\frac{1}{\sqrt{N}}X_N)^d\right] = \sum\limits_{g=0}^\infty N^{-2g} E_g^d,

where E_g^d is the number of ways to glue the edges of a d-gon in pairs so as to produce a compact orientable surface of genus g. The proof is a beautiful piece of mathematics and a nice treatment can be found here. Theorem 14.2 implies Theorem 14.1 because the number of ways to glue a sphere from a polygon with 2k sides is \mathrm{Cat}(k), and only the genus zero term survives in the N \to \infty limit.

There is something unsatisfying about the above sequence of ideas. We know that the entire distribution of X_N is completely determined by the joint distribution of its diagonal matrix elements

X_N(1,1),\dots,X_N(N,N),

which are simply iid standard Gaussians. In fact, at this point in the course we know much more: the spectral analysis of unitarily invariant random Hermitian matrices is equivalent to the analysis of pairs

\begin{matrix} X_{11} & {} & {} \\ X_{21} & X_{22} & {} \\ X_{31} & X_{32} & X_{33} \\ \vdots & \vdots & \vdots \end{matrix}

and

\begin{matrix} B_{11} & {} & {} \\ B_{21} & B_{22} & {} \\ B_{31} & B_{32} & B_{33} \\ \vdots & \vdots & \vdots \end{matrix},

of triangular arrays of real random variables related by

\begin{bmatrix} X_{N1} \\ \vdots \\ X_{NN}\end{bmatrix}=\begin{bmatrix} {} & \vdots & {} \\ \dots & |U_N(i,j)|^2 & \dots \\ {} & \vdots & {} \end{bmatrix}\begin{bmatrix} B_{N1} \\ \vdots \\ B_{NN}\end{bmatrix}

where U_N=[U_N(i,j)] is a uniformly random unitary matrix. Initially, X_{N1},\dots,X_{NN} were viewed as the diagonal matrix elements of a random Hermitian matrix with unitarily invariant distribution and B_{N1},\dots,B_{NN} were the eigenvalues of this selfadjoint matrix. But, we can in fact forget about this original setup entirely, and simply be probabilists thinking about two randomly coupled triangular arrays of real random variables. The question then how to recover Theorem 14.1 simply from the given fact that our X-array has rows X_N=(X_{N1},\dots,X_{NN}) of iid standard Gaussians.

This point of view is why the paper of Olshanski-Vershik is so relevant for our purposes, and so far ahead of its time: they are proving inverse theorems of this kind. More precisely, they are considering the case where the B-array is a given deterministic data set, and proving limit theorems about the corresponding X-array. A special case of their main result is the following “inverse Wigner theorem.”

Theorem 14.3 (Olshanski-Vershik). Suppose that the limit

\gamma_d = \lim\limits_{N \to \infty}\frac{p_d(\left( \frac{B_{N1}}{\sqrt{N}},\dots,\frac{B_{NN}}{\sqrt{N}}\right)}{N}

exists for each d \in \mathbb{N}. Then, for any fixed r \in \mathbb{N}, the random variables X_{N1},\dots,X_{Nr} converge in distribution to an r-tuple Y_1,\dots,Y_r of iid centered Gaussians with variance \gamma_2+\gamma_1^2.

We have proved the one-dimensional version of this Gaussian limit theorem (the case r=1), and we will finish the proof of the multivariate version next time (the case r>1). In our approach to these inverse results of Olshanski-Vershik we are using a combinatorial result which is analogous to Theorem 14.2, but arguably even more miraculous: a universal genus expansion for joint cumulants of the X-array in our theory of unistochastically coupled triangular arrays.

Published by Jonathan Novak

Mathematician at UC San Diego.

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