Gaussian free fields on Hamming graphs and lattice spin systems

TL;DR

This paper studies a class of discrete Gaussian free fields on Hamming graphs, highlighting their distinct properties from traditional lattice spin systems, with explicit results derived using group theory and Fourier analysis.

Contribution

It introduces a new class of Gaussian free fields on Hamming graphs and provides detailed analytical results, contrasting them with classical lattice models.

Findings

Explicit formulas for partition functions and covariances

Distinct interaction structure based on Hamming distance

Analytical results using group theory and Fourier transform

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.