1. Local laws and rigidity for Coulomb gases at any temperature
- Author
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Sylvia Serfaty and Scott N. Armstrong
- Subjects
Statistics and Probability ,Surface (mathematics) ,Superadditivity ,FOS: Physical sciences ,Rigidity (psychology) ,01 natural sciences ,Microscopic scale ,Point process ,010104 statistics & probability ,symbols.namesake ,Subadditivity ,FOS: Mathematics ,Coulomb ,0101 mathematics ,Gibbs measure ,Mathematical Physics ,point process ,Mathematics ,large deviations principle ,Coulomb gas ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,49S05 ,82B05, 60G55, 60F10, 49S05 ,rigidity ,Law ,symbols ,60G55 ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,82B05 ,60F10 - Abstract
We study Coulomb gases in any dimension $d \geq 2$ 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 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 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., Comment: 87 pages, a computational mistake in the proof of Prop. 4.5 corrected. Changes compared to the published version in Annals of Probability are highlighted in color
- Published
- 2021