# 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?

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'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