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The High Temperature Crossover for General 2D Coulomb Gases

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Abstract

We consider N particles in the plane, influenced by a general external potential, that are subject to the Coulomb interaction in two dimensions at inverse temperature \(\beta \). At large temperature, when scaling \(\beta =2c/N\) with some fixed constant \(c>0\), in the large-N limit we observe a crossover from Ginibre’s circular law or its generalisation to the density of non-interacting particles at \(\beta =0\). Using Ward identities and saddle point methods we derive a partial differential equation of generalised Liouville type for the crossover density. For radially symmetric potentials we present some asymptotic results and give examples for the numerical solution of the crossover density. These findings generalise previous results when the interacting particles are confined to the real line. In that situation we derive an integral equation for the resolvent valid for a general potential as well, and present the analytic solution for the density in the case of a Gaussian plus logarithmic potential.

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Notes

  1. For an interpretation of these real and quaternionic Ginibre ensembles as a multi component Coulomb gas we refer to [33].

  2. Note that these and several other authors [9] use a different convention, denoting \(\beta = 2\tilde{\beta }\).

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Acknowledgements

The authors gratefully acknowledge discussions and helpful suggestions of Trinh Khanh Duy, Adrien Hardy, Nam-Gyu Kang, Mylène Maïda, Seong-Mi Seo, Pierpaolo Vivo and Oleg Zaboronski, as well as detailed comments by Yacin Ameur and Gaultier Lambert on a preliminary version of the paper. We also wish to express our gratitude to Jeongho Kim for several valuable comments concerning the numerical verifications.

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  1. Faculty of Physics, Bielefeld University, P.O. Box 100131, 33501, Bielefeld, Germany

    Gernot Akemann

  2. Department of Mathematical Sciences, Seoul National University, Seoul, 151-747, Republic of Korea

    Sung-Soo Byun

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  1. Gernot Akemann
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Correspondence to Gernot Akemann.

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Financial support to Gernot Akemann by the German Research Foundation (DFG) through CRC1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications” and to Sung-Soo Byun by Samsung Science and Technology Foundation (SSTFBA1401-01) are acknowledged. Both authors are equally grateful to the DFG’s International Research Training Group IRTG 2235 supporting the Bielefeld-Seoul exchange programme.

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Akemann, G., Byun, SS. The High Temperature Crossover for General 2D Coulomb Gases. J Stat Phys 175, 1043–1065 (2019). https://doi.org/10.1007/s10955-019-02276-6

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