Mean eigenvector self-overlap in the real and complex elliptic Ginibre ensembles at strong and weak non-Hermiticity
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| Formato: | Preprint |
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2024
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| _version_ | 1866917618816909312 |
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| author | Crumpton, Mark J. Fyodorov, Yan V. Würfel, Tim R. |
| author_facet | Crumpton, Mark J. Fyodorov, Yan V. Würfel, Tim R. |
| contents | We study the mean diagonal overlap of left and right eigenvectors associated with complex eigenvalues in $N\times N$ non-Hermitian random Gaussian matrices. In well known works by Chalker and Mehlig the expectation of this (self-)overlap was computed for the complex Ginibre ensemble as $N\to \infty$. In the present work, we consider the same quantity in the real and complex elliptic Ginibre ensembles characterized by correlations between off-diagonal entries controlled by a parameter $τ\in[0,1]$, with $τ=1$ corresponding to the Hermitian limit. We derive exact expressions for the mean diagonal overlap in both ensembles at any finite $N$, for any eigenvalue off the real axis. We further investigate several scaling regimes as $N\rightarrow \infty$, both in the limit of strong non-Hermiticity keeping a fixed $τ\in[0,1)$ and in the weak non-Hermiticity limit, with $τ$ approaching unity in such a way that $N(1-τ)$ remains finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09296 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mean eigenvector self-overlap in the real and complex elliptic Ginibre ensembles at strong and weak non-Hermiticity Crumpton, Mark J. Fyodorov, Yan V. Würfel, Tim R. Mathematical Physics We study the mean diagonal overlap of left and right eigenvectors associated with complex eigenvalues in $N\times N$ non-Hermitian random Gaussian matrices. In well known works by Chalker and Mehlig the expectation of this (self-)overlap was computed for the complex Ginibre ensemble as $N\to \infty$. In the present work, we consider the same quantity in the real and complex elliptic Ginibre ensembles characterized by correlations between off-diagonal entries controlled by a parameter $τ\in[0,1]$, with $τ=1$ corresponding to the Hermitian limit. We derive exact expressions for the mean diagonal overlap in both ensembles at any finite $N$, for any eigenvalue off the real axis. We further investigate several scaling regimes as $N\rightarrow \infty$, both in the limit of strong non-Hermiticity keeping a fixed $τ\in[0,1)$ and in the weak non-Hermiticity limit, with $τ$ approaching unity in such a way that $N(1-τ)$ remains finite. |
| title | Mean eigenvector self-overlap in the real and complex elliptic Ginibre ensembles at strong and weak non-Hermiticity |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2402.09296 |