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Gaussian free fields on Hamming graphs and lattice spin systems

Shuhei Mano

TL;DR

This paper analyzes discrete Gaussian free fields on the Hamming graph $H(d,n)$ with interactions determined solely by Hamming distance, contrasting them with lattice spin systems. It develops a distance-based random-walk representation and a Fourier/group-theoretic framework to obtain explicit Green functions, partition functions, and covariance structures, covering both massive and massless regimes and boundary conditions. The work then solves three explicit models—distance-independent, nearest-neighbour, and binomially-weighted interactions—providing closed-form expressions for $Z$, $G_{\\alpha,\\partial}$, and covariances across all mass and boundary configurations. Key findings include Gaussian-field universality in large-$d$ or large-$n$ limits, exponential decay of covariances with the Hamming distance, and a sharp divergence of covariances as $m\to 0$ that becomes tempered only after thermodynamic limits. Overall, the paper extends the Gaussian free field framework to highly connected finite graphs, revealing how graph structure dictates fluctuations and correlations through explicit spectral and combinatorial tools.

Abstract

We discuss a class of discrete Gaussian free fields on Hamming graphs, where interactions are determined solely by the Hamming distance between vertices. The purpose of examining this class is that it differs significantly from the commonly discussed spin system on the integer lattice with nearest-neighbour interactions. After introducing general results on the partition function and covariance for the class of Gaussian free fields, we present detailed properties of some specific models. Group-theoretic arguments and the Fourier transform give some explicit results.

Gaussian free fields on Hamming graphs and lattice spin systems

TL;DR

This paper analyzes discrete Gaussian free fields on the Hamming graph with interactions determined solely by Hamming distance, contrasting them with lattice spin systems. It develops a distance-based random-walk representation and a Fourier/group-theoretic framework to obtain explicit Green functions, partition functions, and covariance structures, covering both massive and massless regimes and boundary conditions. The work then solves three explicit models—distance-independent, nearest-neighbour, and binomially-weighted interactions—providing closed-form expressions for , , and covariances across all mass and boundary configurations. Key findings include Gaussian-field universality in large- or large- limits, exponential decay of covariances with the Hamming distance, and a sharp divergence of covariances as that becomes tempered only after thermodynamic limits. Overall, the paper extends the Gaussian free field framework to highly connected finite graphs, revealing how graph structure dictates fluctuations and correlations through explicit spectral and combinatorial tools.

Abstract

We discuss a class of discrete Gaussian free fields on Hamming graphs, where interactions are determined solely by the Hamming distance between vertices. The purpose of examining this class is that it differs significantly from the commonly discussed spin system on the integer lattice with nearest-neighbour interactions. After introducing general results on the partition function and covariance for the class of Gaussian free fields, we present detailed properties of some specific models. Group-theoretic arguments and the Fourier transform give some explicit results.
Paper Structure (17 sections, 19 theorems, 145 equations)

This paper contains 17 sections, 19 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. The random walk representation
  3. The massive without boundary case
  4. The massless with boundary case
  5. The partition function and covariance
  6. The partition function
  7. The covariance
  8. Specific models
  9. The distance-independent interaction
  10. The massive without boundary case
  11. The massive with boundary case
  12. The massless with boundary case
  13. The nearest-neighbour interaction
  14. The massive without boundary case
  15. The massive with boundary case
  16. ...and 2 more sections

Key Result

Proposition 2.1

For given $\mathcal{Y}\subset \mathcal{X}$ and boundary condition $g_x=0$, $x\in\partial$, the values $(\xi_x)_{x\in\mathcal{X}}\in\mathbb{R}^\mathcal{X}$ of the Gaussian free field $(g_x)_{x\in\mathcal{X}}$ satisfy

Theorems & Definitions (38)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4: GM25a
  • Theorem 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 28 more