Gaussian free fields on Hamming graphs and lattice spin systems

We discuss a class of discrete Gaussian free fields on Hamming graphs, where interactions are determined solely by the Hamming distance between vertices. The purpose of examining this class is that it differs significantly from the commonly discussed spin system on the integer lattice with nearest-neighbour interactions. After introducing general results on the partition function and covariance for the class of Gaussian free fields, we present detailed properties of some specific models. Group-theoretic arguments and the Fourier transform give some explicit results.