  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 $\beta$ satisfies $N^{-1} \ll \beta \ll N^{-\frac{1}{2}}$, where
$N$ is the number of particles. This expands the temperature regime for which
convergence to a Poisson point process has been proved.
