We analyze spectral density of ensembles of positive hermitian random matrices related to free convolutions of Marchenko-Pastur distributions. Furthermore, we study asymptotic support of the spectrum and numerical range (field of values) for ensembles of non-hermitian random matrices with independent Gaussian entries.
This is joint work with Özlem Beyarslan. For a fixed finitely generated group G, we consider actions of G by field automorphisms. If the theory of such generic actions is first-order axiomatizable, then we say that G-TCF exists. It is well-known that G-TCF exists if G is a free group (the theory ACFA_n), and it is also known that G-TCF exists for a finite G. On the other hand, it is also known that (Z^2)-TCF does not exist.
Using Bass-Serre theory, we give plausible axioms for G-TCF if G is virtually free. We show that in such a case, the theory G-TCF is simple if and only if G is free or finite.