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).

Proof of the Christoffel–Darboux formula without induction

In this post I prove the Christoffel–Darboux formula without using induction. It seems that often the Christoffel–Darboux formula is proved with induction. However, I find that the proof with induction does not give insight why the Christoffel–Darboux formula is correct. I found the proof below in a paper by Barry Simon.