The field "H" is defined as the long wavelength Fourier components of the spin variable, in the limit that the wavelengths are long. There are many ways to take the long wavelength average, depending on the details of how high wavelengths are cut off. The details are not too important, since the goal is to find the statistics of "H" and not the spins. Once the correlations in "H" are known, the long-distance correlations between the spins will be proportional to the long-distance correlations in "H".