Infinite volume limit for the dipole gas

Here dp is the normalized rotation invariant measure on Sd−1. The fields φ(x) are a family of Gaussian random variables indexed by x ∈ Z with mean zero and covariance given by the positive definite function C(x, y). The measure μC is the underlying measure. To make the connection with the dipole gas one expands the exponential in (4) and carries out the Gaussian integrals. Similarly one can define correlation functions in terms of the field theory.