Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem

We develop an analytic theory of operator-valued additive free convolution in terms of subordination functions. In contrast to earlier investigations our functions are not just given by power series expansions, but are defined as Fréchet analytic functions in all of the operator upper half plane. Furthermore, we do not have to assume that our state is tracial. Combining this new analytic theory of operator-valued free convolution with Anderson’s selfadjoint version of the linearization trick we are able to provide a solution to the following general random matrix problem: Let {X_{1}^{(N)},\dots,X_{n}^{(N)}} be selfadjoint {N\times N} random matrices which are, for {N\to\infty} , asymptotically free. Consider a selfadjoint polynomial p in n non-commuting variables and let {P^{(N)}} be the element {P^{(N)}=p(X_{1}^{(N)},\dots,X_{n}^{(N)})} . How can we calculate the asymptotic eigenvalue distribution of {P^{(N)}} out of the asymptotic eigenvalue distributions of {X_{1}^{(N)},\dots,X_{n}^{(N)}} ?

Discrete Tracy–Widom operators

Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. We consider discrete Tracy–Widom operators and give sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equation and the Fourier transform of Mathieu's equation.