Deformed single ring theorems
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912155157135360 |
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| author | Ho, Ching-Wei Zhong, Ping |
| author_facet | Ho, Ching-Wei Zhong, Ping |
| contents | Given a sequence of deterministic matrices $A = A_N$ and a sequence of deterministic nonnegative matrices $Σ=Σ_N$ such that $A\to a$ and $Σ\to σ$ in $\ast$-distribution for some operators $a$ and $σ$ in a finite von Neumann algebra $\mathcal{A}$. Let $U =U_N$ and $V=V_N$ be independent Haar-distributed unitary matrices. We use free probability techniques to prove that, under mild assumptions, the empirical eigenvalue distribution of $UΣV^*+A$ converges to the Brown measure of $T+a$, where $T\in\mathcal{A}$ is an $R$-diagonal operator freely independent from $a$ and $\vert T\vert$ has the same distribution as $σ$. The assumptions can be removed if $A$ is Hermitian or unitary. By putting $A= 0$, our result removes a regularity assumption in the single ring theorem by Guionnet, Krishnapur and Zeitouni. We also prove a local convergence on optimal scale, extending the local single ring theorem of Bao, Erdős and Schnelli. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_11147 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Deformed single ring theorems Ho, Ching-Wei Zhong, Ping Probability Mathematical Physics Operator Algebras Given a sequence of deterministic matrices $A = A_N$ and a sequence of deterministic nonnegative matrices $Σ=Σ_N$ such that $A\to a$ and $Σ\to σ$ in $\ast$-distribution for some operators $a$ and $σ$ in a finite von Neumann algebra $\mathcal{A}$. Let $U =U_N$ and $V=V_N$ be independent Haar-distributed unitary matrices. We use free probability techniques to prove that, under mild assumptions, the empirical eigenvalue distribution of $UΣV^*+A$ converges to the Brown measure of $T+a$, where $T\in\mathcal{A}$ is an $R$-diagonal operator freely independent from $a$ and $\vert T\vert$ has the same distribution as $σ$. The assumptions can be removed if $A$ is Hermitian or unitary. By putting $A= 0$, our result removes a regularity assumption in the single ring theorem by Guionnet, Krishnapur and Zeitouni. We also prove a local convergence on optimal scale, extending the local single ring theorem of Bao, Erdős and Schnelli. |
| title | Deformed single ring theorems |
| topic | Probability Mathematical Physics Operator Algebras |
| url | https://arxiv.org/abs/2210.11147 |