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Strong asymptotic freeness for independent uniform variables on compact groups associated to non-trivial representations

Charles Bordenave, Benoit Collins

Abstract

Asymptotic freeness of independent Haar distributed unitary matrices was discovered by Voiculescu. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random permutations acting on the orthogonal of the Perron-Frobenius eigenvector. In this paper, we consider a new matrix unitary model appearing naturally from representation theory of compact groups. We fix a non-trivial signature $ρ$, i.e. two finite sequences of non-increasing natural numbers, and for $n$ large enough, consider the irreducible representation $V_{n,ρ}$ of $\mathbb{U}_n$ associated to the signature $ρ$. We consider the quotient $\mathbb{U}_{n,ρ}$ of $\mathbb{U}_n$ viewed as a matrix subgroup of $\mathbb{U}(V_{n,ρ})$, and show that strong asymptotic freeness holds in this generalized context when drawing independent copies of the Haar measure. We also obtain the orthogonal variant of this result. Thanks to classical results in representation theory, this result is closely related to strong asymptotic freeness for tensors, which we establish as a preliminary. In order to achieve this result, we need to develop four new tools, each of independent theoretical interest: (i) a centered Weingarten calculus and uniform estimates thereof, (ii) a systematic and uniform comparison of Gaussian moments and unitary moments of matrices, (iii) a generalized and simplified operator valued non-backtracking theory in a general $C^*$-algebra, and finally, (iv) combinatorics of tensor moment matrices.

Strong asymptotic freeness for independent uniform variables on compact groups associated to non-trivial representations

Abstract

Asymptotic freeness of independent Haar distributed unitary matrices was discovered by Voiculescu. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random permutations acting on the orthogonal of the Perron-Frobenius eigenvector. In this paper, we consider a new matrix unitary model appearing naturally from representation theory of compact groups. We fix a non-trivial signature , i.e. two finite sequences of non-increasing natural numbers, and for large enough, consider the irreducible representation of associated to the signature . We consider the quotient of viewed as a matrix subgroup of , and show that strong asymptotic freeness holds in this generalized context when drawing independent copies of the Haar measure. We also obtain the orthogonal variant of this result. Thanks to classical results in representation theory, this result is closely related to strong asymptotic freeness for tensors, which we establish as a preliminary. In order to achieve this result, we need to develop four new tools, each of independent theoretical interest: (i) a centered Weingarten calculus and uniform estimates thereof, (ii) a systematic and uniform comparison of Gaussian moments and unitary moments of matrices, (iii) a generalized and simplified operator valued non-backtracking theory in a general -algebra, and finally, (iv) combinatorics of tensor moment matrices.

Paper Structure

This paper contains 25 sections, 30 theorems, 198 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Strong asymptotic freeness for random unitary tensors
  4. Main result
  5. Overview of the proof
  6. High order Gaussian approximation for random unitary matrices
  7. Wick calculus
  8. Wick formula
  9. Centered Wick formula
  10. Weingarten calculus with brackets
  11. Unitary Weingarten formula
  12. The centered case
  13. From Weingarten calculus to Wick calculus
  14. Case without brackets
  15. Case with brackets
  16. ...and 10 more sections

Key Result

Proposition 1

The average matrix $\mathbf {E} V$ is the orthogonal projection, denoted by $P_{H}$, onto the vector subspace $H$ of elements invariant under (left) multiplication by $\bar{U} ^ {q_-} \otimes U ^ {q_+}$ for all $U \in \mathbb{U}_n$.

Figures (1)

  • Figure 1: Paired intervals of length $4$. The letters on the lines represent the different brackets that are equal. The first two are paired with color $1$, and the third with color $-1$.

Theorems & Definitions (52)

  • Proposition 1
  • proof : Outline of the proof
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Proposition 6
  • Corollary 7
  • proof
  • Proposition 8
  • ...and 42 more