Energy of the Coulomb gas on the sphere at low temperature

TL;DR

This paper analyzes the energy behavior of Coulomb gas particles on a sphere at low temperature, demonstrating convergence to minimal energy with high probability and average, under specific temperature conditions.

Contribution

It establishes that Coulomb gas configurations on the sphere approach minimal energy at low temperature with high probability and on average, with precise error bounds.

Findings

Logarithmic energy approaches minimal energy at low temperature.

Convergence occurs with exponentially high probability.

Results hold for temperature of order 1/N.

Abstract

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 $\mathcal O(1/N)$.