Convergence for noncommutative rational functions evaluated in random matrices
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....
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| Vydáno v: | Mathematische annalen Ročník 388; číslo 1; s. 543 - 574 |
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01.01.2024
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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
I
I
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. |
|---|---|
| AbstractList | 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 II1 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. 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 I I 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. |
| Author | Miyagawa, Akihiro Yin, Sheng Mai, Tobias Parraud, Félix Collins, Benoît |
| Author_xml | – sequence: 1 givenname: Benoît surname: Collins fullname: Collins, Benoît organization: Department of Mathematics, Kyoto University – sequence: 2 givenname: Tobias surname: Mai fullname: Mai, Tobias organization: Department of Mathematics, Saarland University – sequence: 3 givenname: Akihiro orcidid: 0000-0002-2525-5283 surname: Miyagawa fullname: Miyagawa, Akihiro email: miyagawa.akihiro.43v@st.kyoto-u.ac.jp organization: Department of Mathematics, Kyoto University – sequence: 4 givenname: Félix surname: Parraud fullname: Parraud, Félix organization: Department of Mathematics, Kyoto University, Université de Lyon, ENSL, UMPA – sequence: 5 givenname: Sheng surname: Yin fullname: Yin, Sheng organization: IMT, Université de Toulouse, UPS, Department of Mathematics, Baylor University |
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d
random matrix models, any noncommutative... One of the main applications of free probability is to show that for appropriately chosen independent copies of d random matrix models, any noncommutative... |
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| SubjectTerms | Algebra Approximation Codes Convergence Mathematics Mathematics and Statistics Matrix algebra Operators (mathematics) Polynomials Random variables Rational functions |
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| Title | Convergence for noncommutative rational functions evaluated in random matrices |
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