Correlation functions between singular values and eigenvalues
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914411846828032 |
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| author | Allard, Matthias Kieburg, Mario |
| author_facet | Allard, Matthias Kieburg, Mario |
| contents | Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant complex random matrix ensembles of finite matrix size, we aim at finding the induced probability measure on $j$ eigenvalues and $k$ singular values that we coin $j,k$-point correlation measure. We find an expression for the $1,k$-point correlation measure which simplifies drastically when assuming that the singular values follow a polynomial ensemble, yielding a closed formula in terms of the kernel corresponding to the determinantal point process of the singular value statistics. These expressions simplify even further when the singular values are drawn from a Pólya ensemble and extend known results between the eigenvalue and singular value statistics of the corresponding bi-unitarily invariant ensemble. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19157 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Correlation functions between singular values and eigenvalues Allard, Matthias Kieburg, Mario Probability Mathematical Physics Statistics Theory 60B20 (Primary) 15B52, 43A90, 42B10, 42C05 (Secondary) Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant complex random matrix ensembles of finite matrix size, we aim at finding the induced probability measure on $j$ eigenvalues and $k$ singular values that we coin $j,k$-point correlation measure. We find an expression for the $1,k$-point correlation measure which simplifies drastically when assuming that the singular values follow a polynomial ensemble, yielding a closed formula in terms of the kernel corresponding to the determinantal point process of the singular value statistics. These expressions simplify even further when the singular values are drawn from a Pólya ensemble and extend known results between the eigenvalue and singular value statistics of the corresponding bi-unitarily invariant ensemble. |
| title | Correlation functions between singular values and eigenvalues |
| topic | Probability Mathematical Physics Statistics Theory 60B20 (Primary) 15B52, 43A90, 42B10, 42C05 (Secondary) |
| url | https://arxiv.org/abs/2403.19157 |