Matrix concentration inequalities and free probability

Afonso S. Bandeira, March T. Boedihardjo, Ramon van Handel

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


A central tool in the study of nonhomogeneous random matrices, the noncommutative Khintchine inequality, yields a nonasymptotic bound on the spectral norm of general Gaussian random matrices X= ∑ igiAi where gi are independent standard Gaussian variables and Ai are matrix coefficients. This bound exhibits a logarithmic dependence on dimension that is sharp when the matrices Ai commute, but often proves to be suboptimal in the presence of noncommutativity. In this paper, we develop nonasymptotic bounds on the spectrum of arbitrary Gaussian random matrices that can capture noncommutativity. These bounds quantify the degree to which the spectrum of X is captured by that of a noncommutative model Xfree that arises from free probability theory. This “intrinsic freeness” phenomenon provides a powerful tool for the study of various questions that are outside the reach of classical methods of random matrix theory. Our nonasymptotic bounds are easily applicable in concrete situations, and yield sharp results in examples where the noncommutative Khintchine inequality is suboptimal. When combined with a linearization argument, our bounds imply strong asymptotic freeness for a remarkably general class of Gaussian random matrix models that may be very sparse, have dependent entries, and lack any special symmetries. When combined with a universality principle, our bounds extend beyond the Gaussian setting to general sums of independent random matrices.

Original languageEnglish (US)
Pages (from-to)419-487
Number of pages69
JournalInventiones Mathematicae
Issue number1
StatePublished - Oct 2023

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Free probability
  • Matrix concentration inequalities
  • Random matrices


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