Matrix Concentration Inequalities and Free Probability II. Twosided Bounds and Applications
Abstract
The first paper in this series introduced a new family of nonasymptotic matrix concentration inequalities that sharply capture the spectral properties of very general Gaussian (as well as nonGaussian) random matrices in terms of an associated noncommutative model. These methods achieved matching upper and lower bounds for smooth spectral statistics, but only provided upper bounds for the spectral edges. Here we obtain matching lower bounds for the spectral edges, completing the theory initiated in the first paper. The resulting twosided bounds enable the study of applications that require an exact determination of the spectral edges to leading order, which is fundamentally beyond the reach of classical matrix concentration inequalities. To illustrate their utility, we undertake a detailed study of phase transition phenomena for spectral outliers of nonhomogeneous random matrices.
 Publication:

arXiv eprints
 Pub Date:
 June 2024
 DOI:
 10.48550/arXiv.2406.11453
 arXiv:
 arXiv:2406.11453
 Bibcode:
 2024arXiv240611453B
 Keywords:

 Mathematics  Probability;
 Mathematics  Functional Analysis;
 Mathematics  Operator Algebras;
 60B20;
 60E15;
 46L53;
 46L54;
 15B52
 EPrint:
 47 pages, 5 figures