Free Probability, Random Matrices, and Representations of Non-commutative Rational Functions

Thermalisation for Wigner matrices

“…explicit speed of convergence) or possibly N dependent test functions (like mesoscopic linear statistics) usually requires high (N -dependent) degree for the polynomials that, in turn, are increasingly difficult for the moment method as well as for the analytic subordination method [7]. Thus the extension of the moment method to more general functions has natural limitations, although there is a remarkable recent development for rational functions [24,28,38,14]. The trace of a smooth cut-off function of a polynomial in GUE and deterministic matrices has been analysed via the Master equation and linearization in [23,29] for the purpose of identifying the norm of the polynomial.…”

Berry-Esseen bounds for the multivariate $\mathcal{B}$-free CLT and operator-valued matrices

“…In this section, we are going to give a brief introduction to linearization techniques. What became known in the free probability community as the "linearization trick" [51,22,21,3,4,9,39] turned out some years ago to be used extensively also in other branches of mathematics, ranging from system engineering over automata theory to the theory of non-commutative rings ; see [24,32] and the references collected therein. In fact, linearization works equally well for noncommutative rational functions, but we will restrict here to the case of noncommutative polynomials.…”