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So as I understand it, if you choose a random parameter and fill the entries of a matrix with that parameter, you get a random matrix. I was wondering if there are some general conditions under which a matrix can be both unitary and random such that the unitarity condition is now being satisfied.

I am aware that there are already examples of random unitary matrices which can be chosen from the Haar measure, so perhaps this question is related to the concept of the Haar measure?

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I understand your question as asking for a constructive method to sample uniformly from the unitary group $U(N)$ or orthogonal group $O(N)$, where "uniformly" is understood in the sense of the Haar measure. A simple method starts from an $N\times N$ matrix filled with independent Gaussian random variables [complex for $U(N)$ and real for $O(N)$]. Then orthonormalize the columns via Gram-Schmidt and you're done.

For a more efficient approach, see How to generate random matrices from the classical compact groups by Francesco Mezzadri.

The construction is implemented in Mathematica in the routines circular unitary ensemble for $U(N)$ and circular real ensemble for $O(N)$.
Caution: the socalled circular orthogonal ensemble does not sample from $O(N)$ but from the coset $U(N)/O(N)$.

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  • $\begingroup$ That makes sense but you use independent Gaussian random variables for the matrix, I was wondering if you would take one single random variable as it were (say, all the possible numbers which can be produced by some particular random number generator) and then fill the matrix with every entry being a value which that one variable can take, I take it this is still a random matrix or to be a random matrix does every entry need to correspond to a different independent random variable? $\endgroup$
    – Hollis Williams
    Commented Jun 3, 2019 at 19:57
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    $\begingroup$ you can use a random number generator to fill the matrix, but it is essential that this random number generator samples from a Gaussian distribution; it your random number generator samples from a uniform distribution, you will have to do some postprocessing to convert that into a Gaussian. $\endgroup$
    – Carlo Beenakker
    Commented Jun 3, 2019 at 20:07
  • $\begingroup$ Just checking that I understand, so in some sense it's possible to create a random matrix by just filling it with a random parameter and that will mean all the entries are independent Gaussian random variables (assuming the random parameter takes it values from a Gaussian distribution)? $\endgroup$
    – Hollis Williams
    Commented Jun 9, 2019 at 17:43
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    $\begingroup$ it all depends on what properties of the eigenvalue statistics you are interested in; for example, if you calculate the density of the eigenphases on the unit circle, it will only be uniform for the Gaussian elements, no matter how large the matrix is; on the other hand, local properties (nearest neighbor statistics) will become universal in the limit of a large matrix, but for that you don't even need to take a unitary matrix, just working with a symmetric Gaussian matrix is enough. $\endgroup$
    – Carlo Beenakker
    Commented Jun 23, 2019 at 20:21
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    $\begingroup$ The density should be strictly flat for any N. $\endgroup$
    – Carlo Beenakker
    Commented Jul 21, 2019 at 18:05
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The notion of random unitary matrix usually refers to the Haar measure on the group of unitary matrices; see

https://case.edu/artsci/math/esmeckes/Meckes_SAMSI_Lecture2.pdf

http://emis.ams.org/journals/EJP-ECP/article/download/2551/2345.pdf

] G. W. Anderson, A. Guionnet, and O. Zeitouni, An introduction to random matrices, Cambridge Studies in Advanced Mathematics, vol. 118, Cambridge University Press, Cambridge, 2010. MR-2760897

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