Duality in Random Matrix Theory

I provide examples of a duality between the Gaussian symplectic ensemble and the Gaussian orthogonal ensemble.

Gaussian Symplectic Ensemble

In this post I define the Gaussian Symplectic Ensemble (GSE). Often the GSE is defined using quaternions; here I only use complex numbers. I give an accurate definition of the symplectic group, of self-dual matrices, of the GSE and give an algorithm to draw random matrices from the GSE.

A virial theorem in the Gaussian Unitary Ensemble

In the Gaussian Unitary Ensemble of \( n \times n \) matrices, one can calculate the following expectation value \begin{equation}\label{eq:20151201a} \mathbb{E} \left[ \sum_{ i \neq j} \dfrac{1}{ \left( \lambda_i - \lambda_j \right)^2} \right] = \dfrac{1}{2} n (n-1) \end{equation} with \( \lambda_1 , \ldots, \lambda_n \) the eigenvalues of the random matrix \( H \). I have normalized the GUE such that \( \mathbb{E} [ H_{ij} H_{kl} ] = \delta_{il} \delta_{jk} \). In this blog post, I check \eqref{eq:20151201a} with a Monte Carlo simulation in Mathematica.

Illustration of the Dyson Ornstein-Uhlenbeck process

I define the Dyson Ornstein-Uhlenbeck process as \begin{equation}\label{eq:20151125a} dX_t = -\alpha X_t dt + H \sqrt{dt} \end{equation} with \( \alpha > 0 \) and \( H \) a random matrix from the Gaussian Unitary Ensemble of \( n \times n \) Hermitian matrices. The eigenvalues \( \lambda_i(t) \) of \( X_t \) then have the following dynamics \begin{equation}\label{eq:20151125b} d\lambda_i = -\alpha \lambda_i dt + \sum_{ j \neq i} \frac{1}{\lambda_i - \lambda_j} dt + dB_i \end{equation} where \( B_1, \ldots, B_n \) are independent Brownian processes. In this post I illustrate the process \eqref{eq:20151125b} numerically.

Illustration of Dyson Brownian Motion

The Dyson Brownian motion is defined as \begin{equation}\label{eq:20151124a} X_{t + dt} = X_t + H \sqrt{dt} \end{equation} with \( H \) a random matrix from the Gaussian Unitary Ensemble of \( n \times n \) Hermitian matrices. It is then well-known that the dynamics of the eigenvalues \( \lambda_i(t) \) of \( X_t \) is described by the process \begin{equation}\label{eq:20151124b} d\lambda_i = \sum_{ j \neq i} \frac{1}{\lambda_i - \lambda_j} dt + dB_i \end{equation} where \( B_1, \ldots, B_n \) are independent Brownian processes. In this post I illustrate the process \eqref{eq:20151124b} numerically.

Proof of a determinantal integration formula

While reading about random matrices I encountered the following formula in a blog post by Terence Tao.

If \( K ( x,y) \) is such that

1. \( \int\! dx \ K(x,x) = \alpha \) 
2. \( \int\! dy \ K(x,y) K(y,z) = K(x,z) \)

then \begin{equation}\label{eq:20151118a} \int dx_{n+1} \det_{i,j \le n+1} \left( K(x_i , x_j ) \right) = (\alpha - n) \det_{i,j \le n} \left( K(x_i , x_j ) \right) \end{equation} For simplicity I have written \( \int \) instead of \( \int_{\mathbb{R}} \). This formula is used when calculating n-point functions in the Gaussian Unitary Ensemble (GUE). Tao gives a short proof of \eqref{eq:20151118a} based on induction and the Laplace expansion of determinants. In this post, I give a proof using integration over Grassmann variables. The reason I am interested in this alternative proof is that I want to compress the calculation of n-point functions in the GUE as much as possible.

Spectral Density in the Gaussian Unitary Ensemble

In this post I perform numerical experiments on the spectral density in the Gaussian Unitary Ensemble (GUE).

On the moments of the Gaussian orthogonal ensemble

I am currently reading Mehta's book on random matrices and decided to implement a Mathematica program to calculate expectation values in the Gaussian orthogonal ensemble (GOE). This is the set of symmetric \( n \times n \) matrices \( H \) with probability measure \begin{equation*} P(H) dH = \mathcal{N} \prod_{ i \le j} dH_{ij} \ \exp \left( -\frac{1}{2}\ \mathrm{tr}(H^2)\right) \end{equation*} My program uses recursion based on Wick's theorem (also called Isserlis' theorem according to Wikipedia), and also some rules for summing over indices in \( n \) dimensions. I used ideas from Derevianko's program Wick.m

Some expectation values in the Gaussian orthogonal ensemble

I calculate some expectation values if the probability measure is given by \begin{equation*} P(H) dH = \mathcal{N} \prod_{ i \le j} dH_{ij} \ \exp \left( -\frac{1}{2}\ \mathrm{tr}(H^2)\right) \end{equation*} Hereby are \( H \) symmetric \( n \times n \) matrices and \(\mathcal{N}\) is the normalization factor. This is a special case of the Gaussian orthogonal ensemble (GOE).

Invariance of the Gaussian orthogonal ensemble

On page 17 in his book , Mehta proves the following result about the ensemble of symmetric \( n \times n \) matrices \( H \)

1. If the ensemble is invariant under every transformation \( H \mapsto R H R^T \) with \( R \) an orthogonal matrix
2. and if all components \( H_{ij}, i \le j \) are independent

then the probability measure has the form \begin{equation}\label{eq:20151025e} \prod_{ i \le j} dH_{ij} \ \exp \left( -a\ \mathrm{tr}(H^2) + b\ \mathrm{tr} H + c \right) \end{equation} with \( a, b \) and \(c \) constants.

I prove here the converse, namely, the probability measure \eqref{eq:20151025e} is invariant under transformations \( H \mapsto R H R^T \).

Gaussian orthogonal ensemble

I am currently reading Mehta's book on random matrices (the first edition because it is thinner than the third). I plan to write some blog posts while studying this book. In chapter 2, Metha defines the Gaussian orthogonal ensemble. This is the set of symmetric \( n \times n \) matrices \( H \) with probability density \begin{equation}\label{eq:20151025a} \prod_{ i \le j} dH_{ij} \ \exp \left( -a\ \mathrm{tr}(H^2) + b\ \mathrm{tr} H + c \right) \end{equation} with \( a, b \) and \(c \) constants. It can be calculated that this density function is invariant under transformations \begin{equation}\label{eq:20151025b} H \mapsto R H R^T \end{equation} with \( R \) an orthogonal matrix.

 This is completely equivalent with the vector case. In the vector case the probability density is \begin{equation}\label{eq:20151025c} \prod_{ i} dx_i \ \exp \left( -a\ \sum_i x^2_i + c \right) \end{equation} This density is invariant under rotations \begin{equation}\label{eq:20151025d} x \mapsto R x \end{equation}

One can see that \eqref{eq:20151025d} is the vector representation of the orthogonal group and \eqref{eq:20151025b} is the representation on symmetric matrices. Because symmetric matrices do not form an irreducible representation of the orthogonal group - I can namely subtract the trace - I wonder at this point if one also studies something like ''Gaussian orthogonal ensemble on traceless symmetric matrices''.