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?