Discrete Gaussian free fields on Hamming graphs

Abstract

We discuss a class of discrete Gaussian free fields on Hamming graphs of dimension $d$ with vertex set $\mathcal{X}=\{0,1,\dots ,n-1{\}}^d$, where interactions depend solely on the Hamming distance between vertices. This class provides a setting distinct from the commonly studied fields on the integer lattice with nearest-neighbour interactions. Using spectral and combinatorial properties of Hamming graphs, we show that the field $(g_x{)}_{x\in \mathcal{X}}$ converges weakly to i.i.d.\ Gaussian random variables in the limit as $d\to \infty$ or $n\to \infty$, while the scaled field $(m{g}_x{)}_{x\in \mathcal{X}}$ becomes perfectly correlated in the massless limit $m\to 0$. We then investigate several specific models. For a model in which the interaction weights follow a binomial probability mass function as a function of Hamming distance, with parameter $\gamma \in (0,1-1/n]$, coupling $\gamma$ to $m$ yields a double-scaling limit in which $(m{g}_x{)}_{x\in \mathcal{X}}$ has correlations that are convex in Hamming distance. Remarkably, at every distance, the correlation is exactly the reciprocal of the number of vertices at that distance from a given vertex, attaining its minimum at a distance strictly between $0$ and $d$.

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