Examples of the Stieltjes transformation

I give the definition and two examples of the Stieltjes transformation.

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.