This talk will focus on the fluctuations of a linear spectral statistic around its mean for $P\left(W_N, D_N\right)$ where $P$ is a polynomial, $W_N$ a Wigner matrix and $D_N$ a deterministic diagonal matrix. I will first consider the case when $P\left(W_N,D_N\right)=W_N+D_N$, based on a joint work with M. Février (U. Paris-Saclay). In the general case of $P$ a selfadjoint noncommutative polynomial, I will present results for the special case of the Stieltjes transform, based on a joint work with S. Belinschi (CNRS, U. Toulouse), M. Capitaine (CNRS,U. Toulouse) and M. Février (U. Paris-Saclay).[-]

We consider independent Hermitian heavy-tailed random matrices. Our model includes the Lévy matrices as well as sparse random matrices with O(1) non-zero entries per row. By representing these matrices as weighted graphs, we derive a large deviation principle for key macroscopic observables. Specifically, we focus on the empirical distribution of eigenvalues, the joint neighborhood distribution, and the joint traffic distribution. As an application, we define a notion of microstates entropy for traffic distribution which is additive under free traffic convolution.[-]

Over the last couple of years, it has become evident that matrix-valued semicircular elements establish strong links between free probability theory and noncommutative algebra. Another surprising connection of this kind was found in a recently finished project with Roland Speicher. We have shown that the Fuglede-Kadison determinant of an arbitrary matrix-valued semicircular element is essentially given by the capacity of its associated covariance map. In addition, we have improved a lower bound by Garg, Gurvits, Oliveira, and Widgerson on this capacity, by making it dimension-independent. Besides analytic tools from operator-valued free probability, these are the crucial ingredients in some novel algorithmic solution to the noncommutative Edmonds' problem which we described in collaboration with Johannes Hoffmann. In my talk, I will present our work and provide the background on free probability and noncommutative algebra required for this purpose.[-]