Operator limit of Wigner matrices I
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915336039694336 |
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| author | Banerjee, Debapratim |
| author_facet | Banerjee, Debapratim |
| contents | We consider the Wigner matrix $W_{n}$ of dimension $n \times n$ as $n \to \infty$. The objective of this paper is two folds: first we construct an operator $\mathcal{W}$ on a suitable Hilbert space $\mathcal{H}$ and then define a suitable notion of convergence such that the matrices $W_{n}$ converge in that notion of convergence to $\mathcal{W}$. We further investigate some properties of $\mathcal{W}$ and $\mathcal{H}$. We show that $\mathcal{H}$ is a nontrivial extension of $L^{2}[0,1]$ with respect to the Lebesgue measure and the spectral measure of $\mathcal{W}$ at any function $f \in L^{2}[0,1]$ is almost surely the semicircular law. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23940 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Operator limit of Wigner matrices I Banerjee, Debapratim Probability Mathematical Physics Functional Analysis Statistics Theory We consider the Wigner matrix $W_{n}$ of dimension $n \times n$ as $n \to \infty$. The objective of this paper is two folds: first we construct an operator $\mathcal{W}$ on a suitable Hilbert space $\mathcal{H}$ and then define a suitable notion of convergence such that the matrices $W_{n}$ converge in that notion of convergence to $\mathcal{W}$. We further investigate some properties of $\mathcal{W}$ and $\mathcal{H}$. We show that $\mathcal{H}$ is a nontrivial extension of $L^{2}[0,1]$ with respect to the Lebesgue measure and the spectral measure of $\mathcal{W}$ at any function $f \in L^{2}[0,1]$ is almost surely the semicircular law. |
| title | Operator limit of Wigner matrices I |
| topic | Probability Mathematical Physics Functional Analysis Statistics Theory |
| url | https://arxiv.org/abs/2503.23940 |