Uniqueness of Markov random fields with higher-order dependencies

Abstract

Markov random fields on a countable set $\sf V$ are studied. They are canonically set by a specification $\gamma$, for which the dependence structure is defined by a pre-modification $(h_e)_{e\in {\sf E}}$ -- a consistent family of functions $h_e : S^e\to [0,+\infty)$, where $S$ is a standard Borel space and $\sf E$ is an infinite collection of finite $e\subset {\sf V}$. Different $e$ may contain distinct number of elements, which, in particular, means that the dependence graph ${\sf H}=({\sf V}, {\sf E})$ is a hypergraph. Given $e\in {\sf E}$, let $\delta (e)$ be the logarithmic oscillation of $h_e$. The result of this work is the assertion that the set of all fields $\mathcal{G}(\gamma)$ is a singleton whenever $\delta(e)$ satisfies a condition, a particular version of which can be $\delta(e) \leq \varkappa g(n_{\sf L}(e))$, holding for all $e$ and some $\sf H$-specific $\varkappa\in (0,1)$. Here $g$ is an increasing function, e.g., $g(n) = a+\log n$, and $n_{\sf L}(e)$ is the degree of $e$ in the line-graph ${\sf L}({\sf H})$, which may grow ad infinitum. This uniqueness condition is essentially less restrictive than those based on classical Dobrushin's methods, according to which either of $|e|$, $n_{\sf L}(e)$ and $\delta(e)$ should be globally bounded. We also prove that its fulfilment implies that the unique element of $\mathcal{G}(\gamma)$ is globally Markov.