Tuesday, December 22, 2015

Examples of the Stieltjes transformation

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

Monday, December 14, 2015

Duality in Random Matrix Theory

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

Saturday, December 12, 2015

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.

Thursday, December 3, 2015

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.