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Strong asymptotic freeness of Haar unitaries in quasi-exponential dimensional representations

Michael Magee, Mikael de la Salle

TL;DR

This work proves almost surely strong asymptotic freeness of i.i.d. Haar unitaries U_i^{(n)} when pushed through irreducible U(n) representations π_{k,ℓ} with total size k+ℓ up to n^{A} for any A<1/42–ε, extending previous log-growth results to quasi-exponential dimensions. The authors combine a novel temperedness criterion, a differentiation-with-respect-to-n^{-1} approach, marked-transverse-map topology, and Weingarten calculus for matrix integrals to control word-map traces and operator norms, showing convergence to the free group C*-norm. Key contributions include a general strong-convergence criterion under tempered coefficient families, explicit bounds for matrix coefficients, and corollaries for SU(n) and U(n) settings. The results advance understanding of how limited randomness interacts with large-dimensional unitary representations, with potential implications for operator algebras and non-commutative probability, by providing precise asymptotics and decay rates in a high-dimensional regime. The methodology blends free-group random-walk techniques, topological arguments on surfaces, and sharp analytic estimates to achieve almost-sure strong convergence in a regime previously inaccessible.

Abstract

We prove almost sure strong asymptotic freeness of i.i.d. random unitaries with the following law: sample a Haar unitary matrix of dimension $n$ and then send this unitary into an irreducible representation of $U(n)$. The strong convergence holds as long as the irreducible representation arises from a pair of partitions of total size at most $n^{\frac{1}{42}-\varepsilon}$ and is uniform in this regime. Previously this was known for partitions of total size up to $\asymp\log n/\log\log n$ by a result of Bordenave and Collins.

Strong asymptotic freeness of Haar unitaries in quasi-exponential dimensional representations

TL;DR

This work proves almost surely strong asymptotic freeness of i.i.d. Haar unitaries U_i^{(n)} when pushed through irreducible U(n) representations π_{k,ℓ} with total size k+ℓ up to n^{A} for any A<1/42–ε, extending previous log-growth results to quasi-exponential dimensions. The authors combine a novel temperedness criterion, a differentiation-with-respect-to-n^{-1} approach, marked-transverse-map topology, and Weingarten calculus for matrix integrals to control word-map traces and operator norms, showing convergence to the free group C*-norm. Key contributions include a general strong-convergence criterion under tempered coefficient families, explicit bounds for matrix coefficients, and corollaries for SU(n) and U(n) settings. The results advance understanding of how limited randomness interacts with large-dimensional unitary representations, with potential implications for operator algebras and non-commutative probability, by providing precise asymptotics and decay rates in a high-dimensional regime. The methodology blends free-group random-walk techniques, topological arguments on surfaces, and sharp analytic estimates to achieve almost-sure strong convergence in a regime previously inaccessible.

Abstract

We prove almost sure strong asymptotic freeness of i.i.d. random unitaries with the following law: sample a Haar unitary matrix of dimension and then send this unitary into an irreducible representation of . The strong convergence holds as long as the irreducible representation arises from a pair of partitions of total size at most and is uniform in this regime. Previously this was known for partitions of total size up to by a result of Bordenave and Collins.
Paper Structure (24 sections, 37 theorems, 193 equations, 1 figure)

This paper contains 24 sections, 37 theorems, 193 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation theory of $\mathrm{U}(n)$
  4. Representation theory of $\mathrm{SU}(n)$
  5. Matrix integral results
  6. Markov brothers inequality
  7. A criterion for strong convergence
  8. Bump functions with small Fourier coefficient
  9. Sobolev algebras in the full group $C^{*}$-algebra
  10. Unique trace and exactness
  11. Proof of Proposition \ref{['prop:criterion_for_strong_convergence']} and Proposition \ref{['prop:criterion_for_strong_convergence_coefficients']}
  12. Random walks on free groups
  13. Proper powers
  14. Random walks and tempered functions
  15. Proof of Theorem \ref{['thm:main']} from Theorem \ref{['thm:random-matrix-est']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $U_{1}^{(n)},\ldots,U_{r}^{(n)}$ denote i.i.d. Haar distributed elements of $\mathrm{U}(n)$. For any $A<\frac{1}{42}$ the following holds almost surely. For any non-commutative $*$-polynomialThat is, a polynomial in $r$ non-commuting indeterminates $X_{1},\ldots,X_{r}$ and their formal adjoints where $x_{1},\ldots,x_{r}$ are generators of a free group $\mathbf{F}_{r}$ and the norm on the righ

Figures (1)

  • Figure 11.1: Auxiliary to proof of Proposition \ref{['prop:key-top-prop']}.

Theorems & Definitions (71)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Theorem 3.1
  • Lemma 4.1
  • proof
  • ...and 61 more