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Strong asymptotic freeness for independent uniform variables on compact groups associated to nontrivial representations

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Voiculescu discovered asymptotic freeness of independent Haar-distributed unitary matrices. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random permutations acting on the orthogonal of the Perron-Frobenius eigenvector. In this paper, we consider a new matrix unitary model appearing naturally from representation theory of compact groups. We fix a nontrivial signature \(\rho \), i.e. two finite sequences of non-increasing natural numbers, and for \(n\) large enough, consider the irreducible representation \(V_{n,\rho }\) of \(\mathbb{U}_{n}\) associated with the signature \(\rho \). We consider the quotient \(\mathbb{U}_{n,\rho }\) of \(\mathbb{U}_{n}\) viewed as a matrix subgroup of \(\mathbb{U}(V_{n,\rho })\), and show that strong asymptotic freeness holds in this generalized context when drawing independent copies of the Haar measure. We also obtain the orthogonal variant of this result. Thanks to classical results in representation theory, this result is closely related to strong asymptotic freeness for tensors, which we establish as a preliminary. To achieve this result, we need to develop four new tools, each of independent theoretical interest: (i) a centered Weingarten calculus and uniform estimates thereof, (ii) a systematic and uniform comparison of Gaussian moments and unitary moments of matrices, (iii) a generalized and simplified operator-valued non-backtracking theory in a general \(C^{*}\)-algebra, and finally, (iv) combinatorics of tensor moment matrices.

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References

  1. Anantharaman, N.: Quantum ergodicity on regular graphs. Commun. Math. Phys. 353(2), 633–690 (2017)

    Article  MathSciNet  Google Scholar 

  2. Anderson, G.W.: Convergence of the largest singular value of a polynomial in independent Wigner matrices. Ann. Probab. 41(3B), 2103–2181 (2013)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  4. Aubrun, G., Szarek, S.: Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory. Mathematical Surveys and Monographs, vol. 223. Am. Math. Soc., Providence (2017)

    Google Scholar 

  5. Bordenave, C.: A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts. Ann. Sci. Éc. Norm. Supér. (4) 53(6), 1393–1439 (2020)

    Article  MathSciNet  Google Scholar 

  6. Bordenave, C., Collins, B.: Eigenvalues of random lifts and polynomials of random permutation matrices. Ann. Math. (2) 190(3), 811–875 (2019)

    Article  MathSciNet  Google Scholar 

  7. 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  Google Scholar 

  8. Collins, B., Gaudreau Lamarre, P.Y.: ∗-Freeness in finite tensor products. Adv. Appl. Math. 83, 47–80 (2017)

    Article  MathSciNet  Google Scholar 

  9. Collins, B., Male, C.: The strong asymptotic freeness of Haar and deterministic matrices. Ann. Sci. Éc. Norm. Supér. (4) 47(1), 147–163 (2014)

    Article  MathSciNet  Google Scholar 

  10. Collins, B., Matsumoto, S.: Weingarten calculus via orthogonality relations: new applications. ALEA Lat. Am. J. Probab. Math. Stat. 14(1), 631–656 (2017)

    Article  MathSciNet  Google Scholar 

  11. Collins, B., Nechita, I.: Gaussianization and eigenvalue statistics for random quantum channels (III). Ann. Appl. Probab. 21(3), 1136–1179 (2011)

    Article  MathSciNet  Google Scholar 

  12. 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  Google Scholar 

  13. Friedman, J., Joux, A., Roichman, Y., Stern, J., Tillich, J.P.: The action of a few random permutations on r-tuples and an application to cryptography. In: Annual Symposium on Theoretical Aspects of Computer Science, pp. 375–386. Springer, Berlin (1996)

    Google Scholar 

  14. Füredi, Z., Komlós, J.: The eigenvalues of random symmetric matrices. Combinatorica 1(3), 233–241 (1981)

    Article  MathSciNet  Google Scholar 

  15. González-Guillén, C.E., Palazuelos, C., Villanueva, I.: Euclidean distance between Haar orthogonal and Gaussian matrices. J. Theor. Probab. 31(1), 93–118 (2018)

    Article  MathSciNet  Google Scholar 

  16. Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants. Graduate Texts in Mathematics, vol. 255. Springer, Dordrecht (2009)

    Google Scholar 

  17. Haagerup, U., Thorbjørnsen, S.: A new application of random matrices: \({\mathrm{Ext}}(C^{*}_{\mathrm{red}}(F_{2}))\) is not a group. Ann. Math. (2) 162(2), 711–775 (2005)

    Article  MathSciNet  Google Scholar 

  18. Harrow, A.W.: Quantum expanders from any classical Cayley graph expander. Quantum Inf. Comput. 8(8–9), 715–721 (2008)

    MathSciNet  Google Scholar 

  19. Hastings, M.B.: Random unitaries give quantum expanders. Phys. Rev. A (3) 76(3), 032315 (2007)

    Article  MathSciNet  Google Scholar 

  20. Hastings, M.B., Harrow, A.W.: Classical and quantum tensor product expanders. Quantum Inf. Comput. 9(3–4), 336–360 (2009)

    MathSciNet  Google Scholar 

  21. Janson, S.: Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics, vol. 129. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  22. Jiang, T.: How many entries of a typical orthogonal matrix can be approximated by independent normals? Ann. Probab. 34(4), 1497–1529 (2006)

    Article  MathSciNet  Google Scholar 

  23. Kotani, M., Sunada, T.: Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo 7(1), 7–25 (2000)

    MathSciNet  Google Scholar 

  24. Kowalski, E.: Spectral theory in Hilbert spaces. (2009). https://people.math.ethz.ch/~kowalski/spectral-theory.pdf

  25. Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)

    Article  MathSciNet  Google Scholar 

  26. Margulis, G.A.: Explicit constructions of expanders. Probl. Pereda. Inf. 9(4), 71–80 (1973)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. Mingo, J.A., Speicher, R.: Free Probability and Random Matrices. Fields Institute Monographs, vol. 35. Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto (2017)

    Google Scholar 

  29. Murphy, G.J.: \(C^{*}\)-Algebras and Operator Theory. Academic Press, Boston (1990)

    Google Scholar 

  30. Nica, A.: Asymptotically free families of random unitaries in symmetric groups. Pac. J. Math. 157(2), 295–310 (1993)

    Article  MathSciNet  Google Scholar 

  31. Pisier, G.: A simple proof of a theorem of Kirchberg and related results on \(C^{*}\)-norms. J. Oper. Theory 35(2), 317–335 (1996)

    MathSciNet  Google Scholar 

  32. Pisier, G.: Quantum expanders and geometry of operator spaces. J. Eur. Math. Soc. 16(6), 1183–1219 (2014)

    Article  MathSciNet  Google Scholar 

  33. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Voiculescu, D.: A strengthened asymptotic freeness result for random matrices with applications to free entropy. Int. Math. Res. Not. 1, 41–63 (1998)

    Article  MathSciNet  Google Scholar 

  36. Watanabe, Y., Fukumizu, K.: Graph zeta function in the Bethe free energy and loopy belief propagation. In: Bengio, Y., Schuurmans, D., Lafferty, J., Williams, C., Culotta, A. (eds.) Advances in Neural Information Processing Systems, vol. 22, pp. 2017–2025 (2009)

    Google Scholar 

  37. Želobenko, D.P.: Compact Lie groups and Their Representations. Translations of Mathematical Monographs, vol. 40. Am. Math. Soc., Providence (1973). Translated from the Russian by Israel Program for Scientific Translations

    Google Scholar 

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Acknowledgements

This paper was initiated while the first author was a visiting JSPS scholar at KU in 2018, and he acknowledges the hospitality of JSPS and of Kyoto University. BC was supported by JSPS KAKENHI 17K18734 and 17H04823 and CB by ANR grant ANR-16-CE40-0024. We are indebted to Narutaka Ozawa for drawing our attention to reference [20]. We also are very grateful to the anonymous referees for their efforts in improving the manuscript.

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  1. Institut de Mathématiques de Marseille, CNRS & Aix-Marseille Université, 3 Pl. Victor Hugo, 13331, Marseille Cedex 03, France

    Charles Bordenave

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

    Benoît Collins

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  1. Charles Bordenave
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Bordenave, C., Collins, B. Strong asymptotic freeness for independent uniform variables on compact groups associated to nontrivial representations. Invent. math. (2024). https://doi.org/10.1007/s00222-024-01259-z

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