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ENSPM 2022 Presentation

Title: Random Matrices, Noncommutative Concentration Inequalities, and Free Probability

Abstract: Random matrix theory has enjoyed numerous connections with other areas of mathematics and its applications for many decades. Much of the literature in this area focus on matrices that possess many symmetries, such as matrices with i.i.d. entries, for which precise analytic results and limit theorems have been established. Much less is known about random matrices with arbitrary structure, such as sparse Wigner matrices, or matrices with very correlated entries, despite their importance in many areas of pure and applied mathematics. Matrix Concentration inequalities such as Matrix Bernstein, and the tightly related noncommutative Khintchine inequality of Lust-Piquard and Pisier, provide bounds the spectrum of structured random matrices that have seen countless applications. Unfortunately, they often include suboptimal dimensional factors. In the middle of the 2010’s, Tropp improved the dimensional dependence of this inequality in certain settings by leveraging cancellations due to non-commutativity of the underlying random matrices, giving rise to the question of whether such dependency could be removed.

Based on joint work with March Boedihardjo and Ramon van Handel, more information at arXiv:2108.06312 [math.PR].