Gaussian Fields on a hypercube from Long Range Random Walks

TL;DR

This paper studies Gaussian Free Fields on hypercubes derived from long-range random walks, characterizing their covariance structure, coupling properties, and asymptotic behavior, with connections to spin glass models and mixture distributions.

Contribution

It introduces a novel class of Gaussian fields linked to long-range hypercube random walks, providing their covariance, coupling, and limit theorems, and exploring connections to de Finetti measures and spin glasses.

Findings

Covariance structure characterized by Green functions.

Limit theorem for level set sums as N approaches infinity.

Gaussian process covariance as a mixture of bivariate normal densities.

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.