Spectral Moments of the real Ginibre ensemble
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911827100696576 |
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| author | Byun, Sung-Soo Forrester, Peter J. |
| author_facet | Byun, Sung-Soo Forrester, Peter J. |
| contents | The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large $N$ expansions. These topics are inter-related. For example, a third order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second order difference equation for the general complex moments $M_{2p}^{\rm r}$. The latter are expressed in terms of the ${}_3 F_2$ hypergeometric functions, with a simplification to the ${}_2 F_1$ hypergeometric function possible for $p=0$ and $p=1$, allowing for the large $N$ expansion of these moments to be obtained. The large $N$ expansion involves both integer and half integer powers of $1/N$. The three term recurrence then provides the large $N$ expansion of the full sequence $\{ M_{2p}^{\rm r} \}_{p=0}^\infty$. Fourth and third order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and properties of the large $N$ expansion of these quantities are determined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08896 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spectral Moments of the real Ginibre ensemble Byun, Sung-Soo Forrester, Peter J. Mathematical Physics Probability Primary 60B20, Secondary 33C20 The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large $N$ expansions. These topics are inter-related. For example, a third order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second order difference equation for the general complex moments $M_{2p}^{\rm r}$. The latter are expressed in terms of the ${}_3 F_2$ hypergeometric functions, with a simplification to the ${}_2 F_1$ hypergeometric function possible for $p=0$ and $p=1$, allowing for the large $N$ expansion of these moments to be obtained. The large $N$ expansion involves both integer and half integer powers of $1/N$. The three term recurrence then provides the large $N$ expansion of the full sequence $\{ M_{2p}^{\rm r} \}_{p=0}^\infty$. Fourth and third order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and properties of the large $N$ expansion of these quantities are determined. |
| title | Spectral Moments of the real Ginibre ensemble |
| topic | Mathematical Physics Probability Primary 60B20, Secondary 33C20 |
| url | https://arxiv.org/abs/2312.08896 |