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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?

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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:

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

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  • $\begingroup$ 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? $\endgroup$
    – lnmaurer
    Apr 23 '14 at 19:01
  • $\begingroup$ Are you asking how to do Monte Carlo on a single (one-dimensional) Gaussian distribution? $\endgroup$
    – Qmechanic
    Apr 23 '14 at 19:20
  • $\begingroup$ 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.) $\endgroup$
    – lnmaurer
    Apr 23 '14 at 19:36
  • $\begingroup$ 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. $\endgroup$
    – Qmechanic
    Apr 23 '14 at 19:47

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