Gaussian Fields on a hypercube from Long Range Random Walks

Abstract

We consider a class of Gaussian Free Fields denoted by $(g_x)_{x \in {\cal V}_N}$, where $ {\cal V}_N = \{0,1\}^N$ and $N\in \mathbb{Z}_+$. 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 ${\cal G}_N := \big (g_{x}\big )_{x \in {\cal V}_N}$ 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 ${\cal V}_\infty$. The Green function and a strong representation for $(g_x)$ are characterized by a point process which involves the de Finetti measure of the increments of the random walk. A limit theorem as $N\to \infty$ 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.