The study of the spectrum of random matrices has largely fallen into two distinct categories. For special, typically highly symmetric models of random matrices that arise in areas such as mathematical physics, explicit formulas and specialized techniques provide an extremely precise asymptotic understanding of the spectrum. On the opposite extreme, very general random matrices whose entries admit an essentially arbitrary structure can be investigated using inequalities from operator theory such as the noncommutative Khintchine inequality, but such methods typically only provide very crude information on the spectrum.

In the past few years, however, powerful new tools were discovered to obtain a sharp understanding, to leading order, of the spectral statistics of arbitrarily structured random matrices under surprisingly minimal assumptions. This is made possible by operator-theoretic inequalities that capture to what extent random matrices behave as associated deterministic operators that arise in free probability theory. I will aim to explain these results, some of the mathematics behind them, and how they can be used to answer questions that were previously out of reach. If time permits, I will aim to explain the importance of noncommutative analogues of hypercontractivity in enabling the sharpest forms of such bounds.

The talk is primarily based on joint works with A. Bandeira, M.Boedihardjo, G. Cipolloni, and D. Schroeder.