Scaling Limits of Planar Symplectic Ensembles
We consider various asymptotic scaling limits 𝑁 → ∞ for the 2𝑁 complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, and we obtain limiting expressions for the corresponding k...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211538 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Scaling Limits of Planar Symplectic Ensembles. Gernot Akemann, Sung-Soo Byun and Nam-Gyu Kang. SIGMA 18 (2022), 007, 40 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859645571142254592 |
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| author | Akemann, Gernot Byun, Sung-Soo Kang, Nam-Gyu |
| author_facet | Akemann, Gernot Byun, Sung-Soo Kang, Nam-Gyu |
| citation_txt | Scaling Limits of Planar Symplectic Ensembles. Gernot Akemann, Sung-Soo Byun and Nam-Gyu Kang. SIGMA 18 (2022), 007, 40 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider various asymptotic scaling limits 𝑁 → ∞ for the 2𝑁 complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, and we obtain limiting expressions for the corresponding kernel for different potentials. The first part is devoted to the symplectic Ginibre ensemble with the Gaussian potential. We obtain the asymptotic at the edge of the spectrum in the vicinity of the real line. The unifying form of the kernel allows us to make contact with the bulk scaling along the real line and with the edge scaling away from the real line, where we recover the known determinantal process of the complex Ginibre ensemble. Part two covers ensembles of Mittag-Leffler type with a singularity at the origin. For potentials 𝑄(ζ)=|ζ|²λ − (2c/𝑁)log|ζ|, with λ > 0 and c > −1, the limiting kernel obeys a linear differential equation of fractional order 1/λ at the origin. For integer 𝑚 =1/λ, it can be solved in terms of Mittag-Leffler functions. In the last part, we derive Ward's equation for planar symplectic ensembles for a general class of potentials. It serves as a tool to investigate the Gaussian and singular Mittag-Leffler universality class. This allows us to determine the functional form of all possible limiting kernels (if they exist) that are translation invariant, up to their integration domain.
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| first_indexed | 2026-03-14T13:50:18Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211538 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T13:50:18Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
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| spelling | Akemann, Gernot Byun, Sung-Soo Kang, Nam-Gyu 2026-01-05T12:29:41Z 2022 Scaling Limits of Planar Symplectic Ensembles. Gernot Akemann, Sung-Soo Byun and Nam-Gyu Kang. SIGMA 18 (2022), 007, 40 pages 1815-0659 2020 Mathematics Subject Classification: 60B20; 33C45; 33E12 arXiv:2106.09345 https://nasplib.isofts.kiev.ua/handle/123456789/211538 https://doi.org/10.3842/SIGMA.2022.007 We consider various asymptotic scaling limits 𝑁 → ∞ for the 2𝑁 complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, and we obtain limiting expressions for the corresponding kernel for different potentials. The first part is devoted to the symplectic Ginibre ensemble with the Gaussian potential. We obtain the asymptotic at the edge of the spectrum in the vicinity of the real line. The unifying form of the kernel allows us to make contact with the bulk scaling along the real line and with the edge scaling away from the real line, where we recover the known determinantal process of the complex Ginibre ensemble. Part two covers ensembles of Mittag-Leffler type with a singularity at the origin. For potentials 𝑄(ζ)=|ζ|²λ − (2c/𝑁)log|ζ|, with λ > 0 and c > −1, the limiting kernel obeys a linear differential equation of fractional order 1/λ at the origin. For integer 𝑚 =1/λ, it can be solved in terms of Mittag-Leffler functions. In the last part, we derive Ward's equation for planar symplectic ensembles for a general class of potentials. It serves as a tool to investigate the Gaussian and singular Mittag-Leffler universality class. This allows us to determine the functional form of all possible limiting kernels (if they exist) that are translation invariant, up to their integration domain. It is our pleasure to thank Boris Khoruzhenko for discussions (G.A.) and both Boris Khoruzhenko and Serhii Lysychkin for sharing with us their results [37, 38, 43] prior to publication. The authors are grateful to the DFG-NRF International Research Training Group IRTG 2235 supporting the Bielefeld-Seoul graduate exchange programme. Furthermore, Gernot Akemann was partially supported by the DFG through the grant CRC 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications”. Sung-Soo Byun and Nam-Gyu Kang were partially supported by Samsung Science and Technology Foundation (SSTF-BA1401-51) and by the National Research Foundation of Korea (NRF-2019R1A5A1028324). Nam-Gyu Kang was partially supported by a KIAS Individual Grant (MG058103) at the Korea Institute for Advanced Study. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Scaling Limits of Planar Symplectic Ensembles Article published earlier |
| spellingShingle | Scaling Limits of Planar Symplectic Ensembles Akemann, Gernot Byun, Sung-Soo Kang, Nam-Gyu |
| title | Scaling Limits of Planar Symplectic Ensembles |
| title_full | Scaling Limits of Planar Symplectic Ensembles |
| title_fullStr | Scaling Limits of Planar Symplectic Ensembles |
| title_full_unstemmed | Scaling Limits of Planar Symplectic Ensembles |
| title_short | Scaling Limits of Planar Symplectic Ensembles |
| title_sort | scaling limits of planar symplectic ensembles |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211538 |
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