Eigenvector decorrelation for random matrices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915826185011200 |
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| author | Cipolloni, Giorgio Erdős, László Henheik, Joscha Kolupaiev, Oleksii |
| author_facet | Cipolloni, Giorgio Erdős, László Henheik, Joscha Kolupaiev, Oleksii |
| contents | We study the sensitivity of the eigenvectors of random matrices, showing that even small perturbations make the eigenvectors almost orthogonal. More precisely, we consider two deformed Wigner matrices $W+D_1$, $W+D_2$ and show that their bulk eigenvectors become asymptotically orthogonal as soon as $\mathrm{Tr}(D_1-D_2)^2\gg 1$, or their respective energies are separated on a scale much bigger than the local eigenvalue spacing. Furthermore, we show that quadratic forms of eigenvectors of $W+D_1$, $W+D_2$ with any deterministic matrix $A\in\mathbf{C}^{N\times N}$ in a specific subspace of codimension one are of size $N^{-1/2}$. This proves a generalization of the Eigenstate Thermalization Hypothesis to eigenvectors belonging to two different spectral families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10718 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eigenvector decorrelation for random matrices Cipolloni, Giorgio Erdős, László Henheik, Joscha Kolupaiev, Oleksii Probability Mathematical Physics 60B20, 82C10 We study the sensitivity of the eigenvectors of random matrices, showing that even small perturbations make the eigenvectors almost orthogonal. More precisely, we consider two deformed Wigner matrices $W+D_1$, $W+D_2$ and show that their bulk eigenvectors become asymptotically orthogonal as soon as $\mathrm{Tr}(D_1-D_2)^2\gg 1$, or their respective energies are separated on a scale much bigger than the local eigenvalue spacing. Furthermore, we show that quadratic forms of eigenvectors of $W+D_1$, $W+D_2$ with any deterministic matrix $A\in\mathbf{C}^{N\times N}$ in a specific subspace of codimension one are of size $N^{-1/2}$. This proves a generalization of the Eigenstate Thermalization Hypothesis to eigenvectors belonging to two different spectral families. |
| title | Eigenvector decorrelation for random matrices |
| topic | Probability Mathematical Physics 60B20, 82C10 |
| url | https://arxiv.org/abs/2410.10718 |