Random Matrices, Intrinsic Freeness, and Sharp Non-Asymptotic Inequalities

2025/10/01 by Bandeira, Afonso S.
#Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2510.01021

Abstract

Random matrix theory has played a major role in several areas of pure and applied mathematics, as well as statistics, physics, and computer science. This lecture aims to describe the intrinsic freeness phenomenon and how it provides new easy-to-use sharp non-asymptotic bounds on the spectrum of general random matrices. We will also present a couple of illustrative applications in high dimensional statistical inference. This article accompanies a lecture that will be given by the author at the International Congress of Mathematicians in Philadelphia in the Summer of 2026.

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Related

• Recent developments in non-asymptotic theory of random matrices
• Lecture Notes on "Random Matrices"
• Free probability and random matrices
• Free Probability Theory
• Random Matrices