Convergence of the Laws of Non-Hermitian Sums of Projections
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917881896239104 |
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| author | Zhou, Max Sun |
| author_facet | Zhou, Max Sun |
| contents | We consider the random matrix model $X_n = P_n + i Q_n$, where $P_n$ and $Q_n$ are independently Haar-unitary rotated Hermitian matrices with at most $2$ atoms in their spectra. Let $(M, τ)$ be a tracial von Neumann algebra and let $p, q \in (M, τ)$, where $p$ and $q$ are Hermitian and freely independent. Our main result is the following convergence result: if the law of $P_n$ converges to the law of $p$ and the law of $Q_n$ converges to the law of $q$, then the empirical spectral distributions of the $X_n$ converges to the Brown measure of $X = p + i q$. To prove this, we use the Hermitization technique introduced by Girko, along with the algebraic properties of projections to prove the key estimate. We also prove a converse statement by using the properties of the Brown measure of $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17159 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of the Laws of Non-Hermitian Sums of Projections Zhou, Max Sun Operator Algebras We consider the random matrix model $X_n = P_n + i Q_n$, where $P_n$ and $Q_n$ are independently Haar-unitary rotated Hermitian matrices with at most $2$ atoms in their spectra. Let $(M, τ)$ be a tracial von Neumann algebra and let $p, q \in (M, τ)$, where $p$ and $q$ are Hermitian and freely independent. Our main result is the following convergence result: if the law of $P_n$ converges to the law of $p$ and the law of $Q_n$ converges to the law of $q$, then the empirical spectral distributions of the $X_n$ converges to the Brown measure of $X = p + i q$. To prove this, we use the Hermitization technique introduced by Girko, along with the algebraic properties of projections to prove the key estimate. We also prove a converse statement by using the properties of the Brown measure of $X$. |
| title | Convergence of the Laws of Non-Hermitian Sums of Projections |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2411.17159 |