On the mean field approximation of the Gibbs measure for weakly interacting diffusions

In this talk we will explore when the invariant Gibbs measure of an N -particle system of weakly interacting diffusion is well approximated by the unique minimiser of the mean field energy. We will compute the exact limit of the associated partition function, by re-interpreting the associated integral as the fluctuation of sampling from the mean field limit measure. With this technique we can obtain sharp estimates all the way up to the phase transition, colloquially defined as the loss of analyticity of the partition function with respect to the inverse temperature.