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I believe the relation between deterministic graphs and free probability you mentioned is not something generic. In fact, the main property of your matrix $M$ which makes connection with free probabil …

You can rewrite $$ S=\frac{1}{M}C^{1/2}XX^T(C^{1/2})^T $$ where $X_{.j}\sim \mathcal{N}(\boldsymbol 0, \boldsymbol I)$. It is then classical that the limiting eigenvalue distribution of $S$ (in the …

These three ensembles are hermitian matrices over a (finite dimensional real) field of numbers, and it is known that the only finite dimensional real fields are the real numbers, the complex numbers ( …

Any real normal matrix $M$ can be written as $M=O\,\mathrm{diag}(B_1,\ldots,B_\ell)\,O^t$ where $O$ is orthogonal and where the blocks $B_j$ are either $1\times 1$ real numbers or $2\times2$ matrices …

If I understand correctly, what you are looking for is Lemma 10 (when m=1) in http://arxiv.org/pdf/1210.2681v3.pdf by Elisabeth and Mark Meckes.

If you accept non-commutative random variables as "other random processes $(X_1,\ldots,X_n)$" then, as Jon Bannon's comment suggests, free probability could provide interesting examples [and I don't t …

Just an heuristic answer: The joint eigenvalue distribution of a GOE random matrix is given by $$ \frac{1}{Z_n}\prod_{i<j}|x_i-x_j|\prod_{i=1}^ne^{-x_i^2/2}dx_i. $$ Thus, $E_{GOE}(|\det(y+A)|e^{-x(tr …

By $\|z_1\|$, I understand you mean the maximal modulus of the $z_i$'s. If you are interested in the process of the $\|z_i\|$'s, you have no chance for a determinantal structure since you may have tw …

By symmetry you're looking at the probability that the maximal eigenvalue is smaller than some number. Explicit inequalities for such events can be obtained by using the tridiagonal representation for …

I know that the limiting eigenvalue distribution satisfies a variational principle (see e.g. the joint work with A. Kuijlaars Large Deviations for a Non-Centered Wishart Matrix) from which you may der …