Poisson Statistics for Coulomb Gases at Intermediate Temperature Regimes

We consider the microscopic statistics of a Coulomb gas in $\mathbb{R}^2$ at intermediate temperatures. In particular, we show that the microscopic point process associated to the Coulomb gas converges to a homogeneous Poisson point process at intermediate temperature regimes $\beta N \rightarrow \infty$ and $\beta \sqrt{N} \log N \rightarrow 0$, extending previous results. Our approach relies on a novel quantitative asymptotic description of correlation functions, which is of its own interest.