Extremal process of the zero-average Gaussian Free Field for $d\ge 3$

Abstract: We consider the Gaussian free field on the torus whose covariance kernel is given by the zero-average Green's function. We show that for dimension $d\ge 3$, the extremal point process associated with this field converges weakly to a Poisson random measure. As an immediate corollary, the maxima of the field converges after appropriate centering and scaling to the Gumbel distribution.