# Universality and matrix concentration inequalities

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Random matrices frequently appear in many different fields — physics,

computer science, applied and pure mathematics. Oftentimes the random

matrix of interest will have non-trivial structure — entries that are

dependent and have potentially different means and variances (e.g.

sparse Wigner matrices, matrices corresponding to adjacencies of random

graphs, sample covariance matrices). However, current understanding of

such complicated random matrices remains lacking. In this talk, I will

discuss recent results concerning the spectrum of sums of independent

random matrices with a.s. bounded operator norms. In particular, I will

demonstrate that under some fairly general conditions, such sums will

exhibit the following universality phenomenon — their spectrum will

lie close to that of a Gaussian random matrix with the same mean and

covariance. No special background in random matrix theory will be

necessary for the audience — basic knowledge of probability and linear

algebra are sufficient.

Joint work with Ramon van Handel https://web.math.princeton.edu/~rvan/tuniv220113.pdf