Classical random matrix theory focuses on the study of highly structured models (e.g. Wigner and Wishart matrices) which are presented as a sequence of random matrices defined for every dimension, whose asymptotic (i.e. as the dimension goes to infinity) spectral properties must be understood in detail. However, modern problems in data and computer science require only a coarser understanding of the random matrices in question, but necessitate  nonasymptotic results in settings where the models are less structured and do not necessarily belong  to a prescribed sequence of matrices. 

In this work we provide tools for analyzing the spectral distribution of any self-adjoint noncommutative polynomial evaluated in arbitrary independent unitarily invariant Hermitian random matrices of a fixed dimension. Our results can be interpreted as a quantitative version of Voiculescu's celebrated asymptotic freeness result.  

This is joint work with Chi-Fang Chen and Joel Tropp.