Interlacing Eigenvectors of Large Gaussian Matrices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915035137179648 |
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| author | Attal, Elie Allez, Romain |
| author_facet | Attal, Elie Allez, Romain |
| contents | We consider the eigenvectors of the principal minor of dimension $n< N$ of the Dyson Brownian motion in $\mathbb{R}^{N}$ and investigate their asymptotic overlaps with the eigenvectors of the full matrix in the limit of large dimension. We explicitly compute the limiting rescaled mean squared overlaps in the large $n\,, N$ limit with $n\,/\,N$ tending to a fixed ratio $q\,$, for any initial symmetric matrix $A\,$. This is accomplished using a Burgers-type evolution equation for a specific resolvent. In the GOE case, our formula simplifies, and we identify an eigenvector analogue of the well-known interlacing of eigenvalues. We investigate in particular the case where $A$ has isolated eigenvalues. Our method is based on analysing the eigenvector flow under the Dyson Brownian motion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_17086 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interlacing Eigenvectors of Large Gaussian Matrices Attal, Elie Allez, Romain Probability Statistical Mechanics Mathematical Finance We consider the eigenvectors of the principal minor of dimension $n< N$ of the Dyson Brownian motion in $\mathbb{R}^{N}$ and investigate their asymptotic overlaps with the eigenvectors of the full matrix in the limit of large dimension. We explicitly compute the limiting rescaled mean squared overlaps in the large $n\,, N$ limit with $n\,/\,N$ tending to a fixed ratio $q\,$, for any initial symmetric matrix $A\,$. This is accomplished using a Burgers-type evolution equation for a specific resolvent. In the GOE case, our formula simplifies, and we identify an eigenvector analogue of the well-known interlacing of eigenvalues. We investigate in particular the case where $A$ has isolated eigenvalues. Our method is based on analysing the eigenvector flow under the Dyson Brownian motion. |
| title | Interlacing Eigenvectors of Large Gaussian Matrices |
| topic | Probability Statistical Mechanics Mathematical Finance |
| url | https://arxiv.org/abs/2409.17086 |