Energy of the Coulomb Gas on the Sphere at Low Temperature

We consider the Coulomb gas of N particles on the sphere and show that the logarithmic energy of the configurations approaches the minimal energy up to an error of order log N, with exponentially high probability and on average, provided the temperature is O(1/N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{O}(1/N)}$$\end{document}.