  In this paper we develop a new renormalization group method, which is based
on conditional expectations and harmonic extensions, to study functional
integrals related with 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 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.
