Asymptotic freeness through unitaries generated by polynomials of Wigner matrices
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917614304886784 |
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| author | Parraud, Félix Schnelli, Kevin |
| author_facet | Parraud, Félix Schnelli, Kevin |
| contents | We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on minimizing the assumption of smoothness on those functions while optimizing the error term with respect to $N$, the size of the matrices. As an application, we build on the idea that the long-time Heisenberg evolution associated to Wigner matrices generates asymptotic freeness as first shown in $[9]$. More precisely given $P$ a self-adjoint non-commutative polynomial and $Y^N$ a $d$-tuple of independent Wigner matrices, we prove that the quantum evolution associated to the operator $P(Y^N)$ yields asymptotic freeness for large times. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_02118 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Asymptotic freeness through unitaries generated by polynomials of Wigner matrices Parraud, Félix Schnelli, Kevin Probability Mathematical Physics Operator Algebras We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on minimizing the assumption of smoothness on those functions while optimizing the error term with respect to $N$, the size of the matrices. As an application, we build on the idea that the long-time Heisenberg evolution associated to Wigner matrices generates asymptotic freeness as first shown in $[9]$. More precisely given $P$ a self-adjoint non-commutative polynomial and $Y^N$ a $d$-tuple of independent Wigner matrices, we prove that the quantum evolution associated to the operator $P(Y^N)$ yields asymptotic freeness for large times. |
| title | Asymptotic freeness through unitaries generated by polynomials of Wigner matrices |
| topic | Probability Mathematical Physics Operator Algebras |
| url | https://arxiv.org/abs/2208.02118 |