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Journal of Statistical Physics

, Volume 175, Issue 6, pp 1043–1065 | Cite as

The High Temperature Crossover for General 2D Coulomb Gases

  • Gernot AkemannEmail author
  • Sung-Soo Byun
Article
  • 73 Downloads

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.

Keywords

2D Coulomb gases Normal random matrices High temperature crossover 

Notes

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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Copyright information

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Authors and Affiliations

  1. 1.Faculty of PhysicsBielefeld UniversityBielefeldGermany
  2. 2.Department of Mathematical SciencesSeoul National UniversitySeoulRepublic of Korea

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