Hard edge asymptotics of correlation functions between singular values and eigenvalues
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866914016625950720 |
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| author | Allard, Matthias |
| author_facet | Allard, Matthias |
| contents | Any square complex matrix of size $n\times n$ can be partially characterized by its $n$ eigenvalues and/or $n$ singular values. While no one-to-one correspondence exists between those two kinds of values on a deterministic level, for random complex matrices drawn from a bi-unitarily invariant ensemble, a bijection exists between the underlying singular value ensemble and the corresponding eigenvalue ensemble. This enabled the recent finding of an explicit formula for the joint probability density between $1$ eigenvalue and $k$ singular values, coined $1,k$-point function. We derive here the large $n$ asymptotic of the $1,k$-point function around the origin (hard edge) for a large subclass of bi-unitarily invariant ensembles called polynomial ensembles and its subclass Pólya ensembles. This latter subclass contains all Meijer-G ensembles and, in particular, Muttalib-Borodin ensembles and the classical Wishart-Laguerre (complex Ginibre), Jacobi (truncated unitary), Cauchy-Lorentz ensembles. We show that the latter three ensembles share the same asymptotic of the $1,k$-point function around the origin. In the case of Jacobi ensembles, there exists another hard edge for the singular values, namely the upper edge of their support, which corresponds to a soft edge for the eigenvalue (soft-hard edge). We give the explicit large $n$ asymptotic of the $1,k$-point function around this soft-hard edge. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15765 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hard edge asymptotics of correlation functions between singular values and eigenvalues Allard, Matthias Probability Mathematical Physics Statistics Theory 60B20, 15B52 Any square complex matrix of size $n\times n$ can be partially characterized by its $n$ eigenvalues and/or $n$ singular values. While no one-to-one correspondence exists between those two kinds of values on a deterministic level, for random complex matrices drawn from a bi-unitarily invariant ensemble, a bijection exists between the underlying singular value ensemble and the corresponding eigenvalue ensemble. This enabled the recent finding of an explicit formula for the joint probability density between $1$ eigenvalue and $k$ singular values, coined $1,k$-point function. We derive here the large $n$ asymptotic of the $1,k$-point function around the origin (hard edge) for a large subclass of bi-unitarily invariant ensembles called polynomial ensembles and its subclass Pólya ensembles. This latter subclass contains all Meijer-G ensembles and, in particular, Muttalib-Borodin ensembles and the classical Wishart-Laguerre (complex Ginibre), Jacobi (truncated unitary), Cauchy-Lorentz ensembles. We show that the latter three ensembles share the same asymptotic of the $1,k$-point function around the origin. In the case of Jacobi ensembles, there exists another hard edge for the singular values, namely the upper edge of their support, which corresponds to a soft edge for the eigenvalue (soft-hard edge). We give the explicit large $n$ asymptotic of the $1,k$-point function around this soft-hard edge. |
| title | Hard edge asymptotics of correlation functions between singular values and eigenvalues |
| topic | Probability Mathematical Physics Statistics Theory 60B20, 15B52 |
| url | https://arxiv.org/abs/2501.15765 |