Semiclassical asymptotics of GLN(C) tensor products and quantum random matrices

• Benoît Collins ^[1] ; Jonathan Novak ^[3] ; Piotr Sniady ^[2]
  1. [1] Kyoto University

     Kyoto University

     Kamigyō-ku, Japón

     
  2. [2] Polish Academy of Sciences

     Polish Academy of Sciences

     Warszawa, Polonia

     
  3. [3] UC San Diego

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 24, Nº. 3, 2018, págs. 2571-2623
• Idioma: inglés
• DOI: 10.1007/s00029-017-0387-6
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• Resumen

  • 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 GLN(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, 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, 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 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.