The high temperature crossover for general 2D Coulomb gases

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 generalization to the density of non-interacting particles at $\beta =0$. Using several different methods we derive a partial differential equation of generalized 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 and present the analytic solution for the density in case of a Gaussian plus logarithmic potential.

arXiv comment31 pages, 5 figures

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Cited by in corpus (11)

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• A localization theorem for the planar Coulomb gas in an external field
• Spacing distribution in the 2D Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at non-integer $\beta$
• Edge scaling limit of the spectral radius for random normal matrix ensembles at hard edge
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• Matrix addition and the Dunkl transform at high temperature
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• Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, Circular $\beta$-ensemble and double confluent Heun equation
• Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures