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January 2021 Local laws and rigidity for Coulomb gases at any temperature
Scott Armstrong, Sylvia Serfaty
Ann. Probab. 49(1): 46-121 (January 2021). DOI: 10.1214/20-AOP1445


We study Coulomb gases in any dimension $\mathsf{d}\geq2$ and in a broad temperature regime. We prove local laws on the energy, separation and number of points down to the microscopic scale. These yield the existence of limiting point processes after extraction, generalizing the Ginibre point process for arbitrary temperature and dimension. The local laws come together with a quantitative expansion of the free energy with a new explicit error rate in the case of a uniform background density. These estimates have explicit temperature dependence, allowing to treat regimes of very large or very small temperature, and exhibit a new minimal lengthscale for rigidity and screening, depending on the temperature. They apply as well to energy minimizers (formally zero temperature). The method is based on a bootstrap on scales and reveals the additivity of the energy modulo surface terms, via the introduction of subadditive and superadditive approximate energies.


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Scott Armstrong. Sylvia Serfaty. "Local laws and rigidity for Coulomb gases at any temperature." Ann. Probab. 49 (1) 46 - 121, January 2021.


Received: 1 July 2019; Revised: 1 April 2020; Published: January 2021
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.1214/20-AOP1445

Primary: 49S05, 60F10, 60G55, 82B05

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 1 • January 2021
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