***This event is in-person and open only to Princeton University ID holders**

# Sharp matrix concentration inequalities

What does the spectrum of a random matrix look like when we make no assumption whatsoever about the covariance pattern of its entries? It may appear hopeless at first sight that anything useful can be said at this level of generality. Nonetheless, a set of tools known as "matrix concentration inequalities" makes it possible to estimate certain spectral statistics of very general random matrices up to logarithmic factors in the dimension. The generality and broad applicability of these inequalities have made them a ubiquitous tool in applied mathematics. On the other hand, it is well known that these inequalities fail to yield sharp results for even the simplest random matrix models.

In this talk, I will describe a powerful new class of matrix concentration inequalities that make it possible to achieve optimal results in many situations that are outside the reach of classical methods. These results are easily applicable in concrete situations, and yield detailed nonasymptotic information on the full spectrum of very general random matrices. These new inequalities arise from an unexpected phenomenon: the spectrum of random matrices is accurately captured by the predictions of free probability theory under surprisingly minimal assumptions. Our proofs quantify the notion that it costs little to be free.

The talk is based on joint work with Afonso Bandeira and March Boedihardjo, and with Tatiana Brailovskaya. No prior background on random matrices or free probability will be assumed.