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

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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.

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References

  1. Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  2. Brouwer, P.W., Beenaker, C.W.J.: Diagrammatic method of integration over the unitary group, with applications to quantum transport in mesoscopic systems. J. Math. Phys. 37, 4904–4934 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bufetov, A., Gorin, V.: Representations of classical Lie groups and quantized free convolution. Geom. Funct. Anal. 25(25), 763–814 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bürgisser, P., Ikenmeyer, C.: Deciding positivity of Littlewood–Richardson coefficients. SIAM J. Discrete Math. 4(27), 1639–1681 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biane, P.: Representations of unitary groups and free convolution. Publ. Res. Inst. Math. Sci. 31(1), 63–79 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Biane, P.: Representations of symmetric groups and free probability. Adv. Math. 138(1), 126–181 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Biane, P.: Parking functions of types A and B. Electron. J. Combin. 9(1), 7 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Collins, B., Matsumoto, S., Novak, J.: An invitation to the Weingarten calculus (2017) (in preparation)

  9. Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. 17, 953–982 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Collins, B., Śniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264(3), 773–795 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Collins, B., Śniady, P.: Asymptotic fluctuations of representations of the unitary groups. Preprint arXiv:0911.5546 (2009)

  12. Collins, B., Śniady, P.: Representations of Lie groups and random matrices. Trans. Am. Math. Soc. 361(6), 3269–3287 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Goulden, I.P., Guay-Paquet, M., Novak, J.: On the convergence of monotone Hurwitz generating functions. Ann. Comb. 21, 73–81 (2017)

  14. Goulden, I.P., Guay-Paquet, M., Novak, J.: Toda equations and piecewise polynomiality for mixed double Hurwitz numbers. SIGMA Symmetry Integrability Geom. Methods Appl. 12, 1–10 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Gross, D.J., Taylor, W.: Two-dimensional QCD is a string theory. Nuclear Phys. B 400(1–3), 181–208 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kirillov, A.A.: Lectures on the Orbit Method. Graduate Studies in Mathematics, vol. 64. American Mathematical Society, Providence (2004)

    MATH  Google Scholar 

  17. Knutson, A., Tao, T.: Honeycombs and sums of hermitian matrices. Not. AMS 48, 175–186 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Kuperberg, G.: Random words, quantum statistics, central limits, random matrices. Methods Appl. Anal. 9(1), 99–118 (2002)

    MathSciNet  MATH  Google Scholar 

  19. Matsumoto, S., Novak, J.: Jucys–Murphy elements and unitary matrix integrals. Int. Math. Res. Not. IMRN 2, 362–397 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Morozov, A.: Unitary Integrals and Related Matrix Models. The Oxford Handbook of Random Matrix Theory, pp. 353–375. Oxford University Press, Oxford (2011)

    MATH  Google Scholar 

  21. Mingo, J.A., Speicher, R.: Free Probability and Random Matrices. Fields Institute Publications, Toronto (2016)

    MATH  Google Scholar 

  22. Mingo, J.A., Śniady, P., Speicher, R.: Second order freeness and fluctuations of random matrices. II. Unitary random matrices. Adv. Math. 209(1), 212–240 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Narayanan, H.: On the complexity of computing Kostka numbers and Littlewood–Richardson coefficients. J. Algebraic Comb. 24, 347–354 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Novak, J.I.: Jucys–Murphy elements and the unitary Weingarten function. In: Noncommutative Harmonic Analysis with Applications to Probability II. Banach Center Publication, vol. 89, pp. 231–235. Polish Academy of Scientific Institute and Mathematics, Warsaw (2010)

  25. Novak, J.: Three lectures on free probability. In: Random Matrix Theory, Interacting Particle Systems, and Integrable Systems. Mathematical Sciences Research Institute Publications, vol. 65, pp. 309–383. Cambridge University Press, New York (2014) (with illustrations by Michael LaCroix)

  26. Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, vol. 335. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  27. Novak, J., Śniady, P.: What is...a free cumulant? Not. Am. Math. Soc. 58, 300–301 (2011)

    MathSciNet  MATH  Google Scholar 

  28. Perelomov, A.M., Popov, V.S.: Casimir operators for the classical groups. Dokl. Akad. Nauk SSSR 174, 287–290 (1967)

    MathSciNet  MATH  Google Scholar 

  29. Samuel, S.: \({\rm U}(N)\) integrals, \(1/N\), and the De Wit–’t Hooft anomalies. J. Math. Phys. 21(12), 2695–2703 (1980)

    Article  MathSciNet  Google Scholar 

  30. Shlyakhtenko, D.: Notes on free probability. ArXiv preprint arXiv:math/0504063 (2005)

  31. Stanley, R.P.: Parking functions and noncrossing partitions. Electron J. Comb. 45, R20 (1997)

    MathSciNet  MATH  Google Scholar 

  32. ’t Hooft, G., De Wit, B.: Nonconvergence of the \(1/n\) expansion for SU\((n)\) gauge fields on a lattice. Phys. Lett. 69B, 61–64 (1977)

    MathSciNet  Google Scholar 

  33. Tao, T.: Topics in Random Matrix Theory. Graduate Studies in Mathematics, vol. 132. American Mathematical Society, Providence (2012)

    MATH  Google Scholar 

  34. Voiculescu, D.V., Dykema, K.J., Nica, A.: A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. In: Free Random Variables. CRM Monograph Series, vol. 1. American Mathematical Society, Providence (1992)

  35. Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104(1), 201–220 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  36. Weyl, H.: The Classical Groups. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ (1997). (their invariants and representations, Fifteenth printing, Princeton Paperbacks)

    MATH  Google Scholar 

  37. Xu, F.: A random matrix model from two-dimensional Yang–Mills theory. Commun. Math. Phys. 190(2), 287–307 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  38. Želobenko, D.P.: Compact Lie Groups and Their Representations. American Mathematical Society, Providence (1973). (Translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, vol. 40)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan

    Benoît Collins

  2. CNRS, Paris, France

    Benoît Collins

  3. Department of Mathematics, UC San Diego, 9500 Gilman Drive, La Jolla, CA, 92093-0112, USA

    Jonathan Novak

  4. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956, Warszawa, Poland

    Piotr Śniady

Authors
  1. Benoît Collins
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  2. Jonathan Novak
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  3. Piotr Śniady
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Correspondence to Jonathan Novak.

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To Philippe Biane, for his 55th birthday.

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Collins, B., Novak, J. & Śniady, P. Semiclassical asymptotics of \(\mathbf {GL}_N({\mathbb {C}})\) tensor products and quantum random matrices. Sel. Math. New Ser. 24, 2571–2623 (2018). https://doi.org/10.1007/s00029-017-0387-6

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