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Strong convergence: a short survey

Ramon van Handel

TL;DR

This survey addresses the strong convergence of random matrices toward limiting operator families, motivated by Voiculescu and established through Haagerup–Thorbjørnsen, among others. It surveys limiting models given by free Haar unitaries and free semicirculars, surveys multiple proof strategies (Schwinger–Dyson, moment methods, interpolation, and the polynomial method), and emphasizes the novel intrinsic freeness principle that yields nonasymptotic spectral approximations for messy structured matrices. The work highlights wide-ranging applications in random graphs, hyperbolic geometry, and operator algebras, including results on random lifts, Schreier graphs, hyperbolic surfaces, and Ext not a group, as well as advances enabled by the polynomial method. It also outlines key open problems, notably the reach of strong convergence beyond freeness and the limits of the polynomial method in less regular models, guiding future research directions.

Abstract

A family of random matrices is said to converge strongly to a limiting family of operators if the operator norm of every noncommutative polynomial of the matrices converges to that of the limiting operators. Recent developments surrounding the strong convergence phenomenon have led to new progress on important problems in random graphs, geometry, operator algebras, and applied mathematics. We review classical and recent results in this area, and their applications to various areas of mathematics.

Strong convergence: a short survey

TL;DR

This survey addresses the strong convergence of random matrices toward limiting operator families, motivated by Voiculescu and established through Haagerup–Thorbjørnsen, among others. It surveys limiting models given by free Haar unitaries and free semicirculars, surveys multiple proof strategies (Schwinger–Dyson, moment methods, interpolation, and the polynomial method), and emphasizes the novel intrinsic freeness principle that yields nonasymptotic spectral approximations for messy structured matrices. The work highlights wide-ranging applications in random graphs, hyperbolic geometry, and operator algebras, including results on random lifts, Schreier graphs, hyperbolic surfaces, and Ext not a group, as well as advances enabled by the polynomial method. It also outlines key open problems, notably the reach of strong convergence beyond freeness and the limits of the polynomial method in less regular models, guiding future research directions.

Abstract

A family of random matrices is said to converge strongly to a limiting family of operators if the operator norm of every noncommutative polynomial of the matrices converges to that of the limiting operators. Recent developments surrounding the strong convergence phenomenon have led to new progress on important problems in random graphs, geometry, operator algebras, and applied mathematics. We review classical and recent results in this area, and their applications to various areas of mathematics.
Paper Structure (25 sections, 21 theorems, 46 equations)

This paper contains 25 sections, 21 theorems, 46 equations.

Table of Contents

  1. Introduction.
  2. Strong convergence.
  3. Limiting models.
  4. Strong asymptotic freeness.
  5. Beyond freeness.
  6. The main approaches to strong convergence.
  7. Intrinsic freeness.
  8. Applications.
  9. Random graphs.
  10. Random lifts of graphs.
  11. Random Schreier graphs.
  12. Geometry.
  13. Hyperbolic surfaces.
  14. Minimal surfaces.
  15. Operator algebras.
  16. ...and 10 more sections

Key Result

Theorem 2.2

Let $\boldsymbol{X}^N=(X_1^N,\ldots,X_r^N)$ be i.i.d. GUE/GOE/GSE matrices of dimension $N$, and $\boldsymbol{s}=(s_1,\ldots,s_r)$ be a free semicircular family. Then $\boldsymbol{X}^N$ converges strongly to $\boldsymbol{s}$.

Theorems & Definitions (24)

  • Definition 1.1: Strong convergence
  • Definition 2.1
  • Theorem 2.2: Haagerup--Thorbjørnsen; Schultz
  • Theorem 2.3: Collins--Male
  • Theorem 2.4: Bordenave--Collins
  • Theorem 2.5: Cassidy
  • Theorem 2.6: Louder--Magee
  • Theorem 2.7: Magee--Puder--van Handel
  • Theorem 2.8: Magee--Thomas
  • Theorem 3.1: Haagerup--Thorbjørnsen
  • ...and 14 more