Gaussian fields on a hypercube from long range random walks

We consider a class of Gaussian Free Fields denoted by (gx)x∈VN, where VN={0,1}N and N∈Z+. These fields are related to a general class of N-dimensional random walks on the hypercube, which are killed at a certain rate. The covariance structure of the Gaussian free field is determined by the Green function of these random walks. There exists a coupling such that the Gaussian free fields GN:=(gx)x∈VN form a Markov chain where N is time. If the N entries of the random walk are exchangeable, then the random variables in the Gaussian field can be coupled with spin glass models. A natural choice is to take the increments of the random walk to be from a de Finetti sequence with elements {0, 1}. The random walk is then well defined on V∞. The Green function and a strong representation for (gx) are characterized by a point process which involves the de Finetti measure of the increments of the random walk. A limit theorem as N → ∞ is found for level set sums of the Gaussian free field. In the limit Gaussian process the covariance function is a mixture of a bivariate normal density, with the correlation mixed by a distribution on [−1,1]. We also study a complex Gaussian field which is the transform of the Gaussian process limit.