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Semiclassical asymptotics of \(\mathrm {GL}_N(\mathbb C)\) tensor products and quantum random matrices. (English) Zbl 1404.22032

The Littlewood-Richardson process is a discrete random point process arising from the isotypic decomposition of tensor products of irreducible representations of the linear group \(\mathrm{GL}_N (\mathbb C)\). Biane-Perelomov-Popov matrices are quantum random matrices obtained as the geometric quantization of random Hermitian matrices with deterministic eigenvalues and uniformly random eigenvectors. By an observation made by Ph. Biane, the correlation functions of certain global observables of the LR process coincide with the correlation functions of linear statistics of sums of classically independent BPP matrices. Furthermore, Biane suggested that the Littlewood-Richardson process should be viewed as quantizing a certain continuous random point process. This gave rise to a random matrix approach to the statistical study of \(\mathrm{GL}_N (\mathbb C)\) tensor products.
In the paper under review, the authors prove that classically independent BPP matrices become freely independent in any semiclassical/large-dimension limit. The result proves and generalizes a conjecture by A. Bufetov and V. Gorin [Geom. Funct. Anal. 25, No. 3, 763–814 (2015; Zbl 1326.22012)], and leads to a law of large numbers for the BPP observables of the LR process which holds in all semiclassical scalings.

MSC:

22E46 Semisimple Lie groups and their representations
60B20 Random matrices (probabilistic aspects)
46L54 Free probability and free operator algebras
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators

Citations:

Zbl 1326.22012
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References:

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