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Partition Functions of Determinantal and Pfaffian Coulomb Gases with Radially Symmetric Potentials

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Abstract

We consider random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian structures, respectively. For a class of radially symmetric potentials with soft edges, we derive the asymptotic expansions of the log-partition functions up to and including the O(1)-terms as the number N of particles increases. Notably, our findings stress that the formulas of the \(O(\log N)\)- and O(1)-terms in these expansions depend on the connectivity of the droplet. For random normal matrix ensembles, our formulas agree with the predictions proposed by Zabrodin and Wiegmann up to an additive constant depending on N but not on the background potential. For planar symplectic ensembles, the expansions contain a new kind of ingredient in the O(N)-terms, the logarithmic potential evaluated at the origin in addition to the entropy of the ensembles.

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Notes

  1. We mention that the physical entropy is \(-\int _{\mathbb {C}} \log (\Delta Q/\pi ) \, d\mu _{Q}\).

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Acknowledgements

We thank Gernot Akemann, Christophe Charlier, Peter Forrester and Thomas Leblé for helpful discussions.

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Authors and Affiliations

  1. Center for Mathematical Challenges, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Republic of Korea

    Sung-Soo Byun

  2. School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Republic of Korea

    Nam-Gyu Kang

  3. Department of Mathematics, Chungnam National University, 99 Daehak-ro, Yuseong-gu, Daejeon, 34134, Republic of Korea

    Seong-Mi Seo

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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). Sung-Soo Byun was partially supported by a KIAS Individual Grant (SP083201) via the Center for Mathematical Challenges at Korea Institute for Advanced Study. Nam-Gyu Kang was partially supported by a KIAS Individual Grant (MG058103) at Korea Institute for Advanced Study. Seong-Mi Seo was partially supported by the National Research Foundation of Korea (NRF-2019R1A5A1028324, NRF-2022R1I1A1A01072052).

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Byun, SS., Kang, NG. & Seo, SM. Partition Functions of Determinantal and Pfaffian Coulomb Gases with Radially Symmetric Potentials. Commun. Math. Phys. 401, 1627–1663 (2023). https://doi.org/10.1007/s00220-023-04673-1

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