Emergence of a Poisson process in weakly interacting particle systems

We consider the Gibbs measure of a general interacting particle system for a certain class of ``weakly interacting" kernels. In particular, we show that the local point process converges to a Poisson point process as long as the inverse temperature β satisfies N⁻¹ β N-1/2, where N is the number of particles. This expands the temperature regime for which convergence to a Poisson point process has been proved.