A Class of Gaussian Fields on $\mathbb{Z}_q^d$

TL;DR

This paper constructs and analyzes a class of Gaussian fields on discrete and continuous spaces derived from long-range random walks, exploring their properties, transformations, and associated limit theorems.

Contribution

It introduces a novel class of Gaussian fields linked to long-range random walks, including their decomposition, transformations, and limit behaviors on various spaces.

Findings

Gaussian fields are constructed from long-range random walks with killing.

A decomposition into independent Gaussian variables is provided.

Limit theorems for the Gaussian fields and partition functions are established.

Abstract

Gaussian fields $(g_x)$ on $\mathbb{Z}_q^d$ are constructed from a class of reversible long range random walks $(X_t)_{t\in \mathbb{N}}$ on $\mathbb{Z}_q^d$ in arXiv:2510.22554. The construction is from taking the covariance function of $(g_x)$ as $(1-\alpha)G(x,y;\alpha)$, where $G(x,y;\alpha)$ is the Green function of a random walk with killing in each transition at rate $1-\alpha$. A decomposition of the Gaussian field into a sum of independent Gaussian random variables is made. By letting $q\to \infty$ the Gaussian field becomes defined from an infinite-dimensional random walk on a torus. The random walk model is also extended to $d=\infty$ by considering a de Finetti random walk where entries in the increments of the random walk are exchangeable. A limit Gaussian field on $\mathbb{R}^d$ arises from a central limit theorem approach. The transform of this Gaussian field, which is again a Gaussian field, is calculated. It has a simpler covariance matrix than the original field. The Hamiltonian connected to the Gaussian field is calculated. A limit theorem for the partition function arising from the Hamiltonian is found.