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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Published in:	Mathematische annalen Vol. 388; no. 1; pp. 543 - 574
Main Authors:	Collins, Benoît, Mai, Tobias, Miyagawa, Akihiro, Parraud, Félix, Yin, Sheng
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2024 Springer Nature B.V
Subjects:	Algebra Approximation Codes Convergence Mathematics Mathematics and Statistics Matrix algebra Operators (mathematics) Polynomials Random variables Rational functions
ISSN:	0025-5831, 1432-1807
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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
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Snippet	One of the main applications of free probability is to show that for appropriately chosen independent copies of 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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