Ramon van Handel is an associate professor, and a member of the executive committee of the Program in Applied and Computational Mathematics (PACM), at Princeton University. He received his PhD in 2007 from Caltech. His honors include the NSF CAREER award, the Presidential Early Career Award for Scientists and Engineers (PECASE), the Erlang Prize, the Princeton University Graduate Mentoring Award, and several teaching awards. He has served on the editorial boards of the Annals of Applied Probability and Probability Theory and Related Fields, and is the probability editor for the IMS Textbooks and Monographs series. Ramon’s research interests lie broadly in probability theory and its interactions with other areas of mathematics. He is particularly fascinated by the development of probabilistic principles and methods that explain the common structure in a variety of pure and applied mathematical problems. His recent interests are focused on high-dimensional phenomena in probability, analysis, and geometry. He has also worked on conditional phenomena in probability and ergodic theory, and on applications of noncommutative probability. This Medallion lecture was given at the Seminar on Stochastic Processes in March 2022. 


Nonasymptotic random matrix theory

Classical random matrix theory is largely concerned with the spectral properties of special models of random matrices, such as matrices with i.i.d. entries or invariant ensembles, whose asymptotic behavior as the dimension increases has been understood in striking detail. On the other hand, suppose we are given a random matrix with an essentially arbitrary pattern of entry means and variances, dependencies, and distributions. What can we say about its spectrum? Beside lacking most of the special features that facilitate the analysis of classical random matrix models, such questions are inherently nonasymptotic in nature: when we are asked to study the spectral properties of a given, arbitrarily structured random matrix, there is no associated sequence of models of increasing dimension that enables us to formulate asymptotic questions.

It may appear hopeless that anything useful can be proved at this level of generality. Nonetheless, a set of tools known as “matrix concentration inequalities” makes it possible at least to crudely estimate the range of the spectrum of very general random matrices up to logarithmic factors in the dimension. Due to their versatility and ease of use, these inequalities have had a considerable impact on a wide variety of applications in pure mathematics, applied mathematics, and statistics. On the other hand, it is well known that these inequalities fail to capture the correct order of magnitude of the spectrum even in the simplest examples of random matrices. Until very recently, results of this kind were essentially the only available tool for the study of generally structured random matrices.

In my lecture, I describe a new approach to such questions (developed in joint work with Bandeira, Boedihardjo, and Brailovskaya) that has opened the door to a drastically improved nonasymptotic understanding of the spectral properties of generally structured random matrices. In this new theory, we introduce certain deterministic infinite-dimensional operators, constructed using methods of free probablity, that may be viewed a the “Platonic ideals” associated random matrices. Our theory shows, in a precise nonasymptotic sense, that the spectrum of an arbitrarily structured random matrix is accurately captured by that of the associated Platonic ideal under remarkably mild conditions. The resulting sharp inequalities are easily applicable in concrete situations, and capture the correct behavior of many examples for which no other approach is known.

[Editor’s note: an accidentally abbreviated version of this preview appears in the August 2022 issue of the IMS Bulletin; the full version will be reprinted in the September issue. Apologies for any confusion caused.]