# Student Probability Seminar

#### Concentration inequality around the thermal equilibrium measure of Coulomb gases

Coulomb gases are described by an ensemble of particles that interact with each other via a Coulombian repulsion and are constrained by a confining potential. We study a temperature regime that stands in the middle of two regimes previously studied: it is a regime in which the temperature is strong enough that no structure is observed at the microscopic level, but weak enough that the mass remains confined in a compact set. Our main result is that, with probability exponentially close to $1,$ the thermal equilibrium measure is the best possible description of the distribution of the system i.e. the distance between the empirical measure and the thermal equilibrium measure is of order $N^{\frac{1}{d}},$ where $N$ is the number of particles and $d$ is the dimension of the ambient space. The thermal equilibrium measure minimizes the free energy functional, which consists of the original hamiltonian, plus an entropy term weighted by the inverse temperature.