# Local laws and rigidity for Coulomb gases at any temperature

## TL;DR

This paper establishes local laws, rigidity, and limiting point processes for Coulomb gases across all temperatures and dimensions, providing explicit error rates and minimal lengthscales that depend on temperature.

## Contribution

It introduces a new method based on scale bootstrap and energy additivity to analyze Coulomb gases, extending results to all temperatures and dimensions.

## Key findings

- Proves local laws for energy, separation, and point counts at microscopic scales.
- Establishes existence of generalized limiting point processes for arbitrary temperature and dimension.
- Provides explicit temperature-dependent error bounds and minimal lengthscales for rigidity.

## 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.