A Renormalization Group Method by Harmonic Extensions and the Classical Dipole Gas

In this paper, we develop a new renormalization group method, which is based on conditional expectations and harmonic extensions, to study functional integrals of small perturbations of Gaussian fields. In this new method, one integrates Gaussian fields inside domains at all scales conditioning on the fields outside these domains, and by the variation principle solves local elliptic problems. It does not rely on an a priori decomposition of the Gaussian covariance. We apply this method to the model of classical dipole gas on the lattice, and show that the scaling limit of the generating function with smooth test functions is the generating function of the renormalized Gaussian free field.