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This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural inverse-distance repulsion between particles." Rather, I'm interested in how to characterize GOE/GUE through their projections, symmetries and invariances.

For a random symmetric matrix of size $n\times n$, with entries drawn from a Gaussian Ensemble, the joint probability of eigenvalues can be written as:

$$\rho_{\beta,n}(x_1,\ldots,x_n)=C_{\beta,n}\prod_{1\leq i<j\leq N}|x_j-x_i|^\beta\prod_{i=1}^ne^{-\beta x_i^2/2}, \ \beta=1,2,3,4\ldots$$

where $\beta$ depends on the specific type of ensemble ($\beta=1$ for GOE, $\beta=2$ for GUE, etc.).

Now, suppose that $X_i$ $(i=1,2,\ldots,n)$ is a sequence of i.i.d. random variables such that the joint distribution of the vector $(X_1,\ldots,X_n)$ is invariant under rotations. Then the vector must be Gaussian.

Here's another characterization of Gaussians: If $Z$ is Gaussian and $X,Y$ are independent r.v. such that $Z=X+Y$, then $X,Y$ must be Gaussian. This is also known as Cramer's theorem.

There are many more characterizations of Gaussians.

Question: Are there some nice characterizations of the $\rho_{\beta,n}$ distributions? I would be particularly interested in the case of GOE ($\beta=1$). Moreover, I would love to see characterizations relying on symmetries, invariance or projections. If these properties exist, have there been usage of these techniques in the literature to show a certain distribution is from a Gaussian Ensemble?

Edit: I should add here that the characterizations I'm looking for should preferably not be related to characterizations of the random matrix ensemble involved. For example the GOE has a distribution that's invariant under orthogonal similarity transformations. I say this because in some problems, a connection to random matrix theory is nonobvious but numerically one sees the statistics are the same.

Here is an example. Take $n$ Brownian bridges starting at $0$ at $t=0$ and ending at $0$ at $t=2$. Now condition them to not intersect. Then the distribution of particle locations at $t=1$ is exactly the same as the GUE eigenvalue distribution. Now, of course there are well known ways of connecting these bridges directly to random matrix theory through the notion of determinental proceeses. However, suppose I didn't know of an immediate connection. Could I somehow convince myself that the GUE distribution is a good guess for this process?

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    $\begingroup$ If I recall, the only models for which there are any known group symmetries are $\beta = 1,2,4$ corresponding to orthogonal, unitary, and symplectic transformations. Last I heard, Chris Sinclair was trying to show that the square integer processes are hyperpfaffian. I think all that is really known about the rest of those processes is that there exists a matrix model for which that is the eigenvalue process. The general $\beta$ ensembles are not really physical unless you interpret them as log-gasses with Coulomb interaction at inverse temperature $\beta$ in a Gaussian potential. $\endgroup$
    – Chris Janjigian
    Commented Mar 13, 2014 at 20:40

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to answer your first question, I could just follow your line of argument and take an ensemble of $n\times n$ Hermitian matrices $A$, with real ($\beta=1$), complex ($\beta=2$) or real quaternion matrix elements ($\beta=4$). If I am told that the distribution of $A$ is invariant under orthogonal ($\beta=1$), unitary ($\beta=2$), or symplectic ($\beta=4$) transformations, and that the numbers $a_{kl}=(1+\delta_{kl})^{1/2}A_{kl}$ ($1\leq k\leq l\leq n$) are i.i.d. random variables, then the eigenvalues $x_1,x_2,\ldots x_n$ of $cA$ have your distribution $\rho_{\beta,n}$ for some $c\geq 0$.

your second question: is this useful --- this characterization is at the basis of algorithms to generate random numbers $x_1,x_2,\ldots x_n$ with the distribution $\rho_{\beta,n}$, which would be very cumbersome to generate directly.

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  • $\begingroup$ Thanks, the second part is particularly enlightening. I've added a slight edit to the question. Basically, is there a non-random matrix characterization of these distributions? $\endgroup$
    – Alex R.
    Commented Mar 11, 2014 at 15:33
  • $\begingroup$ non-random matrix characterizations: $\rho_{\beta,n}$ is the Gibbs distribution of a one-dimensional Coulomb gas in a harmonic confining potential, at temperature $1/\beta$, but that does not single out the special values $\beta=1,2,4$; any such characterization will somehow have to involve the Jacobian from matrix elements to eigenvalues, how could this arise otherwise? $\endgroup$
    – Carlo Beenakker
    Commented Mar 11, 2014 at 16:32
  • $\begingroup$ That's a great question. I'm not entirely if what I'm asking for is reasonable. I've added an example below my edit which I think encapsulates what I'm after. $\endgroup$
    – Alex R.
    Commented Mar 11, 2014 at 19:25
  • $\begingroup$ Carlo's Gibbs measure+ requirement to be determinantal characterizes, I believe but have not double checked, $\beta=2$. The $\beta=1$ and $\beta=4$ can be characterized by decimation relations. $\endgroup$
    – ofer zeitouni
    Commented Mar 11, 2014 at 20:16

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