Uniqueness of Markov random fields with higher-order dependencies

Markov random fields on a countable set $\mathsf{V}$ are studied. They are canonically set by a specification $\gamma$, for which the dependence structure is defined by a pre-modification $(h_e{)}_{e\in \mathsf{E}}$ – a consistent family of functions $h_e:S^e\to [0,+\mathrm{\infty })$, where $S$ is a standard Borel space and $\mathsf{E}$ is an infinite collection of finite $e\subset \mathsf{V}$. Different $e$ may contain distinct number of elements, which, in particular, means that the dependence graph $\mathsf{H}=(\mathsf{V},\mathsf{E})$ is a hypergraph. Given $e\in \mathsf{E}$, 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 $\delta (e)\le \mathrm{\varkappa }g(n_{\mathsf{L}}(e))$, holding for all $e$ and some $\mathsf{H}$-specific $\mathrm{\varkappa }\in (0,1)$. Here $g$ is an increasing function, e.g., $g(n)=a+\log n$, and $n_{\mathsf{L}}(e)$ is the degree of $e$ in the line-graph $\mathsf{L}(\mathsf{H})$, 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 $\mathrm{\mid }e\mathrm{\mid }$, $n_{\mathsf{L}}(e)$ 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.