Harer-Zagier type recursion formula for the elliptic GinOE
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916260375166976 |
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| author | Byun, Sung-Soo |
| author_facet | Byun, Sung-Soo |
| contents | We consider the real eigenvalues of the elliptic Ginibre matrix indexed by the non-Hermiticity parameter $τ\in [0,1]$, and present a Harer-Zagier type recursion formula for the even moments in the form of an $11$-term recurrence relation. For the symmetric GOE case ($τ=1$), it reduces to a known 5-term recurrence relation. On the other hand, for the asymmetric cases when $τ< 1$, the recursion formula is new, even in the special case of the well-studied Ginibre ensemble ($τ=0$), where it reduces to a 3-term recurrence. For the proof, we derive a seventh-order linear differential equation for the moment generating function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_11185 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Harer-Zagier type recursion formula for the elliptic GinOE Byun, Sung-Soo Probability Mathematical Physics We consider the real eigenvalues of the elliptic Ginibre matrix indexed by the non-Hermiticity parameter $τ\in [0,1]$, and present a Harer-Zagier type recursion formula for the even moments in the form of an $11$-term recurrence relation. For the symmetric GOE case ($τ=1$), it reduces to a known 5-term recurrence relation. On the other hand, for the asymmetric cases when $τ< 1$, the recursion formula is new, even in the special case of the well-studied Ginibre ensemble ($τ=0$), where it reduces to a 3-term recurrence. For the proof, we derive a seventh-order linear differential equation for the moment generating function. |
| title | Harer-Zagier type recursion formula for the elliptic GinOE |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2309.11185 |