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Convergence for noncommutative rational functions evaluated in random matrices

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Abstract

One of the main applications of free probability is to show that for appropriately chosen independent copies of d random matrix models, any noncommutative polynomial in these d variables has a spectral distribution that converges asymptotically and can be described with the help of free probability. This paper aims to show that this can be extended to noncommutative rational functions, answering an open question by Roland Speicher. This paper also provides a noncommutative probability approach to approximating the free field. At the algebraic level, its construction relies on the approximation by generic matrices. On the other hand, it admits many embeddings in the algebra of operators affiliated with a \(II_1\) factor. A consequence of our result is that, as soon as the generators admit a random matrix model, the approximation of any self-adjoint noncommutative rational function by generic matrices can be upgraded at the level of convergence in distribution.

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Acknowledgements

The problem considered in this paper appeared in the context of the M.Sc. studies of A. Miyagawa, under the supervision of B. Collins (cf. [34]), and related questions were discussed during the visit of T. Mai in Kyoto in 2019. T. Mai is grateful for the great hospitality of B. Collins and the entire Department of Mathematics at Kyoto University. F. Parraud, T. Mai, and S. Yin benefited from the hospitality of MFO, during which R. Speicher stated the conjecture leading to this paper. The authors thank G. Cébron, A. Connes, M. de la Salle, and R. Speicher for valuable discussions. We thank an anonymous referee for a careful reading of our manuscript and for valuable comments and suggestions. BC was supported by JSPS KAKENHI 17K18734 and 17H04823. FP was partially supported by Labex Milyon (ANR-10-LABX-0070) of Université de Lyon. SY was supported by ANR project MESA (ANR-18-CE40-0006).

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Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan

    Benoît Collins, Akihiro Miyagawa & Félix Parraud

  2. Department of Mathematics, Saarland University, 66123, Saarbrücken, Germany

    Tobias Mai

  3. Université de Lyon, ENSL, UMPA, 46 allée d’Italie, 69007, Lyon, France

    Félix Parraud

  4. IMT, Université de Toulouse, UPS, 31400, Toulouse, France

    Sheng Yin

  5. Department of Mathematics, Baylor University, Waco, TX, 76706, USA

    Sheng Yin

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  1. Benoît Collins
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  3. Akihiro Miyagawa
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Correspondence to Akihiro Miyagawa.

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Collins, B., Mai, T., Miyagawa, A. et al. Convergence for noncommutative rational functions evaluated in random matrices. Math. Ann. 388, 543–574 (2024). https://doi.org/10.1007/s00208-022-02530-5

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