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There is a lot of literature on random matrices, however, in most of the sources that I have seen, the standard construction is by iid sampling of elements of the matrix. While it is natural from mathematical point of view I do not see practical interest in such construction: if an operator models some physical process, hardly it can be related by iid sampling of its elements in some specific coordinate system.

  • Could someone tell me in which applications random matrices with iid sampling make a really reasonable model? (I guess that in some areas iid sampling could be reasonable - like graphs/network/social modelling - but I am not sure).

From Bayesian inference it is known that distributions on some classes of matrices are possible, like the Wishart distribution on positive definite matrices. For me it is the opposite side of iid sampled matrices (which I find very "unstructured") - these are too "structured" and very fast there is a limit in modelling by standard tools (instead of priors we start to use mixtures which are often barely interpretable (I mean the mixing parameters)) ...

  • I wonder if there is something in between - e.g. I start from some operator (matrix) which is deterministic and reasonable for modelling and slowly increase its complexity by adding randomizations (I prefer term 'jittering') that do not break initial physical meaning.
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    $\begingroup$ In random matrix theory it is more common to consider models like the Gaussian unitary ensemble, etc., where the entries will not all be iid because of restrictions that the matrices be hermitian, etc. $\endgroup$
    – Sam Hopkins
    Commented 19 hours ago
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    $\begingroup$ For the last point, the simplest approach is to take the operator and add a random Hermitian operator. This is described by the Dyson Brownian motion process. $\endgroup$
    – Will Sawin
    Commented 17 hours ago
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    $\begingroup$ @WillSawin But what it means to add random Hermitian operator? At which point this could be interesting? $\endgroup$
    – Fedor Goncharov
    Commented 15 hours ago
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    $\begingroup$ Oh I just mean take a matrix where the entires are iid Gaussian except for the condition that they be Hermitian, as Sam Hopkins said, and add it to your matrix. This seems like possibly the simplest model of 'jittering' a Hermitian matrix, and might be interesting for that reason, or see the literature on Dyson Brownian motion for reasons people have found this interesting in the past. $\endgroup$
    – Will Sawin
    Commented 15 hours ago
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    $\begingroup$ Concerning physical applications, it's hard to find a treatment of quantum chaos that doesn't mention random matrix theory. The spectral densities of random matrix ensembles behave like those of quantum chaotic systems. $\endgroup$
    – Michael Engelhardt
    Commented 12 hours ago

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Q: Are there any physical processes from the real world that are modeled by truly random operators that have their components iid / or not iid (in some basis)?

A widely studied example of a random matrix with components that are not i.i.d. is localization of waves by a random potential (Anderson localization). The Hamiltonian is then a banded random matrix, with the nonzero matrix elements clustered along the principal diagonal. Of course, this representation is then basis dependent, which is the essence of localization: eigenstates are confined to nearby lattice sites.
For an overview, see Random band matrices by Paul Bourgade.

Conversely, a system where the eigenstates are extended rather than localized is well described by a Hamiltonian that is statistically equivalent in any basis, which leads to the representation of i.i.d. Gaussian matrix elements.

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