Optimal Delocalization for Non--Hermitian Eigenvectors
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912592914546688 |
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| author | Cipolloni, Giorgio Landon, Benjamin |
| author_facet | Cipolloni, Giorgio Landon, Benjamin |
| contents | We prove an optimal order delocalization estimate for the eigenvectors of general $N \times N$ non-Hermitian matrices $X$: $\| {\bf v } \|_\infty \leq C \sqrt{\frac{\log N}{N}}$ with very high probability, for any right or left eigenvector ${\bf v}$ of $X$. This improves upon the previous tightest bound of Rudelson and Vershynin [arXiv:1306.2887] of $\mathcal{O}( ( \log N)^{9/2}N^{-1/2})$, and holds under weaker assumptions on the tail of the matrix elements. In addition to the coordinate basis, our bound holds for the $\ell^\infty$ norm in any deterministic orthonormal basis. Our result is proven via a dynamical method, by studying the flow of the resolvent of the Hermitization of $X$ and proving local laws on short scales. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15189 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Delocalization for Non--Hermitian Eigenvectors Cipolloni, Giorgio Landon, Benjamin Probability Mathematical Physics We prove an optimal order delocalization estimate for the eigenvectors of general $N \times N$ non-Hermitian matrices $X$: $\| {\bf v } \|_\infty \leq C \sqrt{\frac{\log N}{N}}$ with very high probability, for any right or left eigenvector ${\bf v}$ of $X$. This improves upon the previous tightest bound of Rudelson and Vershynin [arXiv:1306.2887] of $\mathcal{O}( ( \log N)^{9/2}N^{-1/2})$, and holds under weaker assumptions on the tail of the matrix elements. In addition to the coordinate basis, our bound holds for the $\ell^\infty$ norm in any deterministic orthonormal basis. Our result is proven via a dynamical method, by studying the flow of the resolvent of the Hermitization of $X$ and proving local laws on short scales. |
| title | Optimal Delocalization for Non--Hermitian Eigenvectors |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2509.15189 |