Distribution of the Diagonal Entries of the Resolvent of a Complex Ginibre Matrix
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917851264188416 |
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| author | Bousseyroux, Pierre Bouchaud, Jean-Philippe Potters, Marc |
| author_facet | Bousseyroux, Pierre Bouchaud, Jean-Philippe Potters, Marc |
| contents | The study of eigenvalue distributions in random matrix theory is often conducted by analyzing the resolvent matrix $ \mathbf{G}_{\mathbf{M}}^N(z) = (z \mathbf{1} - \mathbf{M})^{-1} $. The normalized trace of the resolvent, known as the Stieltjes transform $ \mathfrak{g}_{\mathbf{M}}^N(z) $, converges to a limit $ \mathfrak{g}_{\mathbf{M}}(z) $ as the matrix dimension $ N $ grows, which provides the eigenvalue density $ ρ_{\mathbf{M}} $ in the large-$ N $ limit. In the Hermitian case, the distribution of $ \mathfrak{g}_{\mathbf{M}}^N(z) $, now regarded as a random variable, is explicitly known when $ z $ lies within the limiting spectrum, and it coincides with the distribution of any diagonal entry of $ \mathbf{G}_{\mathbf{M}}^N(z) $.
In this paper, we investigate what becomes of these results when $ \mathbf{M} $ is non-Hermitian. Our main result is the exact computation of the diagonal elements of $ \mathbf{G}_{\mathbf{M}}^N(z) $ when $ \mathbf{M} $ is a Ginibre matrix of size $ N $, as well as the high-dimensional limit for different regimes of $ z $, revealing a tail behavior connected to the statistics of the left and right eigenvectors. Interestingly, the limit distribution is stable under inversion, a property previously observed in the symmetric case. We then propose two general conjectures regarding the distribution of the diagonal elements of the resolvent and its normalized trace in the non-Hermitian case, both of which reveal a symmetry under inversion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19266 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distribution of the Diagonal Entries of the Resolvent of a Complex Ginibre Matrix Bousseyroux, Pierre Bouchaud, Jean-Philippe Potters, Marc Mathematical Physics Disordered Systems and Neural Networks Probability The study of eigenvalue distributions in random matrix theory is often conducted by analyzing the resolvent matrix $ \mathbf{G}_{\mathbf{M}}^N(z) = (z \mathbf{1} - \mathbf{M})^{-1} $. The normalized trace of the resolvent, known as the Stieltjes transform $ \mathfrak{g}_{\mathbf{M}}^N(z) $, converges to a limit $ \mathfrak{g}_{\mathbf{M}}(z) $ as the matrix dimension $ N $ grows, which provides the eigenvalue density $ ρ_{\mathbf{M}} $ in the large-$ N $ limit. In the Hermitian case, the distribution of $ \mathfrak{g}_{\mathbf{M}}^N(z) $, now regarded as a random variable, is explicitly known when $ z $ lies within the limiting spectrum, and it coincides with the distribution of any diagonal entry of $ \mathbf{G}_{\mathbf{M}}^N(z) $. In this paper, we investigate what becomes of these results when $ \mathbf{M} $ is non-Hermitian. Our main result is the exact computation of the diagonal elements of $ \mathbf{G}_{\mathbf{M}}^N(z) $ when $ \mathbf{M} $ is a Ginibre matrix of size $ N $, as well as the high-dimensional limit for different regimes of $ z $, revealing a tail behavior connected to the statistics of the left and right eigenvectors. Interestingly, the limit distribution is stable under inversion, a property previously observed in the symmetric case. We then propose two general conjectures regarding the distribution of the diagonal elements of the resolvent and its normalized trace in the non-Hermitian case, both of which reveal a symmetry under inversion. |
| title | Distribution of the Diagonal Entries of the Resolvent of a Complex Ginibre Matrix |
| topic | Mathematical Physics Disordered Systems and Neural Networks Probability |
| url | https://arxiv.org/abs/2411.19266 |