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

Robert Griffiths, Shuhei Mano

TL;DR

This paper constructs and analyzes a broad class of Gaussian fields indexed by ${\cal V}_{q,d}$ arising from reversible long-range random walks on ${\cal Z}_q^d$ with killing parameter $\alpha$. It develops spectral decompositions via circulant structures, extends to continuous-state tori and infinite dimensions via de Finetti exchangeability, and establishes central limit limits to Gaussian fields on $\mathbb{R}^d$; it also derives a transform with a simpler covariance and a Hamiltonian leading to a partition-function limit. The work unifies Green-function/killing techniques, de Finetti mixtures, and multivariate Krawtchouk polynomials to obtain explicit eigenstructures for both discrete and continuous models, including torus and infinite-dimensional settings. It further connects these Gaussian fields to Hamiltonians and partition-functions, and discusses Potts-model-type interpretations and free-energy implications in large dimension. The results provide a versatile framework for analyzing Gaussian fields linked to long-range random-walk structures and their thermodynamic limits, with potential applications to spin-glass-type phenomena on discrete and continuous spaces.

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 $(1-α)G(x,y;α)$, where $G(x,y;α)$ is the Green function of a random walk with killing in each transition at rate $1-α$. 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.

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

TL;DR

This paper constructs and analyzes a broad class of Gaussian fields indexed by arising from reversible long-range random walks on with killing parameter . It develops spectral decompositions via circulant structures, extends to continuous-state tori and infinite dimensions via de Finetti exchangeability, and establishes central limit limits to Gaussian fields on ; it also derives a transform with a simpler covariance and a Hamiltonian leading to a partition-function limit. The work unifies Green-function/killing techniques, de Finetti mixtures, and multivariate Krawtchouk polynomials to obtain explicit eigenstructures for both discrete and continuous models, including torus and infinite-dimensional settings. It further connects these Gaussian fields to Hamiltonians and partition-functions, and discusses Potts-model-type interpretations and free-energy implications in large dimension. The results provide a versatile framework for analyzing Gaussian fields linked to long-range random-walk structures and their thermodynamic limits, with potential applications to spin-glass-type phenomena on discrete and continuous spaces.

Abstract

Gaussian fields on are constructed from a class of reversible long range random walks on in arXiv:2510.22554. The construction is from taking the covariance function of as , where is the Green function of a random walk with killing in each transition at rate . A decomposition of the Gaussian field into a sum of independent Gaussian random variables is made. By letting the Gaussian field becomes defined from an infinite-dimensional random walk on a torus. The random walk model is also extended to by considering a de Finetti random walk where entries in the increments of the random walk are exchangeable. A limit Gaussian field on 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.
Paper Structure (16 sections, 22 theorems, 151 equations)

This paper contains 16 sections, 22 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Stochastic processes from circulants
  3. Continuous time processes
  4. The Green function and killing
  5. de Finetti exchangeability
  6. Random walk on an infinite-dimensional torus
  7. Gaussian free fields
  8. Gaussian field on ${\cal V}_{q,{\textcolor{black}{d}}}$
  9. de Finetti Gaussian field
  10. Exchangeable entries in $V$ and spin glasses
  11. Gaussian field on a torus
  12. Central limit form for the Green function as ${\textcolor{black}{d}}\to \infty$
  13. Gaussian field of a transform
  14. Gaussian Field and Hamiltonians
  15. Hamiltonian
  16. ...and 1 more sections

Key Result

Proposition 1

(GM2025a.) Let $x,y \in {\cal V}_{{\textcolor{black}{q}} ,{\textcolor{black}{d}}}$. Then is non-negative and therefore a transition probability matrix if and only if $\rho_r = \mathbb{E} [\theta_1^{V{\hbox{\large\bfseries .}} r} ]$ for a random variable $V$ on ${\cal V}_{{\textcolor{black}{q}} ,{\textcolor{black}{d}}}$. The stationary distribution is uniform in ${\cal V}_{{\textcolor{b

Theorems & Definitions (55)

  • Proposition 1
  • Remark 1
  • Example 1
  • Proposition 2
  • Proposition 3
  • Corollary 1
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • ...and 45 more