# What do matrices in the Gaussian orthogonal ensemble look like?

I've been reading a fair amount about quantum chaos, and random matrix theory comes up a lot. I get that they're looking at the distribution of eigenvalues from an ensemble of random matrices, but I still don't know what the ensemble of those matrices looks like.

For example, take the Gaussian orthogonal ensemble. If I understand correctly, the matrices are real and symmetric, and when generating such an ensemble, the probability of a given $n\times n$ matrix $H$ being generated is proportional to $e^{-\frac{n}{4}trH^2}$. How would I generate I generate such an ensemble (on a computer)? What is the distribution of the individual random elements of the matrix? Gaussian?

## 1 Answer

It is not just a distribution of eigenvalues. The underlying distribution for the Gaussian orthogonal ensemble GOE(n) of real symmetric $n\times n$ matrices is $\frac{n(n+1)}{2}$ dimensional, and given in terms of the $\frac{n(n+1)}{2}$ independent matrix elements, each with a Gaussian weight factor:

$$\left[\prod_{i,j\in\{1,\ldots, n\}}^{i\leq j} \int_{\mathbb{R}}\mathrm{d}H_{ij}\right] e^{-\frac{1}{2}{\rm tr}(H^2)}(\ldots) ~=~ \left[\prod_{i\in\{1,\ldots, n\}} \int_{\mathbb{R}}\mathrm{d}H_{ii}~e^{-\frac{1}{2}H_{ii}^2}\right] \left[\prod_{i,j\in\{1,\ldots, n\}}^{i<j} \int_{\mathbb{R}}\mathrm{d}H_{ij}~e^{-H_{ij}^2}\right] (\ldots)$$

• I guess I wasn't clear: for the applications I'm interested in, the relevant part is the distribution of their eigenvalues; I'm not claiming that the GOE is a distribution of eigenvalues. If I wanted to generate an ensemble, could I generate a bunch of random, orthogonal matrices and do rejection technique on them using that weight factor? Does that weight factor say anything intuitive about the distribution of the individual matrix elements? Apr 23 '14 at 19:01
• Are you asking how to do Monte Carlo on a single (one-dimensional) Gaussian distribution? Apr 23 '14 at 19:20
• Maybe? I know how to generate a bunch of random numbers according to a gaussian distribution. I'd like to do likewise with these matrices. (I see GOE repeated all over the place, but I want to actually see a small ensemble of some darn matrices.) Apr 23 '14 at 19:36
• I'm not entirely sure what you are asking, but it seems that you just have to do Monte Carlo on a product of $\frac{n(n+1)}{2}$ independent (one-dimensional) Gaussian distributions. Apr 23 '14 at 19:47