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Selecta Mathematica

, Volume 24, Issue 3, pp 2571–2623 | Cite as

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

  • Benoît Collins
  • Jonathan Novak
  • Piotr Śniady
Article
  • 40 Downloads

Abstract

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. 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 \(\mathrm {GL}_N({\mathbb {C}})\) 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.

Keywords

Asymptotic representation theory Random matrix theory Free probability Representations of general linear groups Quantization 

Mathematics Subject Classification

Primary 22E46 Secondary 60B20 46L54 34L20 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceKyoto UniversityKyotoJapan
  2. 2.CNRSParisFrance
  3. 3.Department of MathematicsUC San DiegoLa JollaUSA
  4. 4.Institute of MathematicsPolish Academy of SciencesWarszawaPoland

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