  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.
