The strong convergence phenomenon

TL;DR

This survey reviews recent advances in strong convergence of random matrices, highlighting new methods that extend classical results and enable applications in operator algebras, graphs, and geometry.

Contribution

It introduces recent methods for establishing strong convergence in more general settings and discusses their powerful applications across various mathematical fields.

Findings

Extended strong convergence results for random matrices.

New quantitative methods for analyzing convergence.

Applications to open problems in operator algebras and geometry.

Abstract

In a seminal 2005 paper, Haagerup and Thorbj{\o}rnsen discovered that the norm of any noncommutative polynomial of independent complex Gaussian random matrices converges to that of a limiting family of operators that arises from Voiculescu's free probability theory. In recent years, new methods have made it possible to establish such strong convergence properties in much more general situations, and to obtain even more powerful quantitative forms of the strong convergence phenomenon. These, in turn, have led to a number of spectacular applications to long-standing open problems on random graphs, hyperbolic surfaces, and operator algebras, and have provided flexible new tools that enable the study of random matrices in unexpected generality. This survey aims to provide an introduction to this circle of ideas.