[Paper Review] Asymptotic fluctuations of representations of the unitary groups

Benoı̂t Collins, Piotr Śniady|arXiv (Cornell University)|Nov 30, 2009

Random Matrices and Applications12 references3 citations

TL;DR

This paper establishes that asymptotic fluctuations of irreducible representations of the unitary group U(n) converge to those of large random matrices under a scaling limit, extending Biane's free probability results beyond mean values to include higher-order fluctuations via higher-order free probability theory. The key contribution is showing that the spectral measures of representations and random matrices exhibit identical Gaussian fluctuations with covariance decay 1/n² as n→∞.

ABSTRACT

We study asymptotics of representations of the unitary groups U(n) in the limit as n tends to infinity and we show that in many aspects they behave like large random matrices. In particular, we prove that the highest weight of a random irreducible component in the Kronecker tensor product of two irreducible representations behaves asymptotically in the same way as the spectrum of the sum of two large random matrices with prescribed eigenvalues. This agreement happens not only on the level of the mean values (and thus can be described within Voiculescu's free probability theory) but also on the level of fluctuations (and thus can be described within the framework of higher order free probability).

Motivation & Objective

• To understand the asymptotic behavior of irreducible representations of U(n) as n→∞, particularly in the context of Kronecker tensor products.
• To extend Biane's earlier result—linking representation theory and free probability—beyond mean values to include fluctuations around the mean.
• To unify the asymptotic behavior of representations of U(n) with that of large random matrices by showing equivalence in both macroscopic and microscopic fluctuations.
• To weaken the technical assumptions on the growth rate of highest weights used in prior work, making the results more general.
• To provide a conceptual explanation for why representations and random matrices exhibit similar asymptotic behavior using non-commutative random matrix theory and convergence in higher-order free probability.

Proposed method

• Associate to each irreducible representation ρ of U(n) a random matrix X(ρ) = U diag(l₁,…,lₙ) U⁻¹, where U is Haar-distributed on U(n), called the 'naïve random matrix'.
• Show that the non-commutativity of the matrix entries in X(ρ) asymptotically vanishes as n→∞, allowing it to be treated as a classical random matrix in the limit.
• Use higher-order free probability theory to describe fluctuations of spectral moments, including cumulants of order ≥3, which characterize Gaussian fluctuations.
• Prove convergence in both macroscopic and microscopic senses of higher-order free probability for the rescaled spectral measures of representations and associated random matrices.
• Apply additivity of cumulants and the Borel-Cantelli lemma to establish almost-sure convergence of spectral moments to their limiting values.
• Use the fact that the spectral measure of the upper-left mₙ×mₙ submatrix of X(ρ) converges to the free compression of the limiting spectral measure μ, linking restriction to subgroups with free probability.

Experimental results

Research questions

• RQ1 Do the asymptotic fluctuations of irreducible representations of U(n) match those of large random matrices beyond just mean values?
• RQ2 Can higher-order free probability theory be used to describe the joint fluctuations of spectral moments of representations and random matrices?
• RQ3 What is the precise scaling limit of the spectral measure of a random irreducible component in the Kronecker product of two representations?
• RQ4 How does the convergence of spectral measures of representations compare to that of the associated naïve random matrices in the limit n→∞?
• RQ5 Can the technical assumptions on the growth of highest weights in Biane's original result be relaxed while preserving the asymptotic equivalence with random matrices?

Key findings

• The spectral measures of irreducible representations of U(n) and their associated naïve random matrices exhibit identical asymptotic fluctuations in the limit n→∞.
• The rescaled spectral measures of representations and random matrices converge almost surely to the same limit, with fluctuations decaying as 1/n².
• The limits of the first and second cumulants of spectral moments match between representations and random matrices, confirming agreement in mean and variance.
• Cumulants of order i≥3 vanish in the limit for both representations and random matrices, indicating Gaussian fluctuations with covariance decay 1/n².
• The convergence of the rescaled spectral measures occurs in both the macroscopic and microscopic senses of higher-order free probability theory.
• The restriction of a representation to a subgroup corresponds to the free compression of the limiting spectral measure, which matches the spectral measure of the upper-left submatrix of the associated random matrix.