Semiclassical asymptotics of $\operatorname{GL}_N(\mathbb{C})$ tensor products and quantum random matrices

TL;DR

This paper establishes that Biane--Perelomov--Popov matrices become freely independent in the large-dimension limit, linking quantum random matrices to semiclassical asymptotics and advancing understanding of tensor product representations of .

Contribution

It proves that BPP matrices are freely independent in the semiclassical limit, confirming a conjecture and extending the connection between quantum matrices and classical asymptotics.

Findings

BPP matrices become freely independent in large dimensions.

A Law of Large Numbers for BPP observables is established.

The results generalize previous conjectures in quantum random matrix theory.

Abstract

The Littlewood--Richardson process is a discrete random point process arising from the isotypic decomposition of tensor products of irreducible representations of $\operatorname{GL}_N(\mathbb{C})$. Biane--Perelomov--Popov matrices are quantum random matrices obtained as the geometric quantization of random Hermitian matrices with deterministic eigenvalues and uniformly random eigenvectors. As first observed by Biane, correlation functions of certain global observables of the LR process coincide with correlation functions of linear statistics of sums of classically independent BPP matrices, thereby enabling a random matrix approach to the statistical study of $\operatorname{GL}_N(\mathbb{C})$ tensor products. In this paper, we prove an optimal result: classically independent BPP matrices become freely independent in any semiclassical/large-dimension limit. This proves and generalizes a conjecture of Bufetov and Gorin, and leads to a Law of Large Numbers for the BPP observables of the LR process which holds in any and all semiclassical scalings.