Semiclassical Asymptotics of Gl N ( C ) Tensor Products and Quantum Random Matrices

The Littlewood–Richardson process is a discrete random point process which encodes the isotypic decomposition of tensor products of irreducible rational representations of GLN (C). Biane-PerelomovPopov matrices are a family of quantum random matrices arising 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 GLN (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 removes all assumptions on the decay rate of the semiclassical parameter present in previous works, and may be viewed as a maximally robust geometric quantization of Voiculescu’s theorem on the asymptotic freeness of independent unitarily invariant random Hermitian matrices. In particular, our work proves and generalizes a conjecture of Bufetov and Gorin, and shows that the mean global asymptotics of GLN (C) tensor products are governed by free probability in any and all semiclassical scalings. Our approach extends to global fluctuations, and thus yields a Law of Large Numbers for the LR process valid in all semiclassical scalings. To Philippe Biane, for his 55th birthday. 2010 Mathematics Subject Classification. 22E46 (Primary) 60B20, 46L54, 34L20 (Secondary) .