Translation invariant extensions of finite volume measures

We investigate the following questions: Given a measure $μ_Λ$ on configurations on a subset $Λ$ of a lattice $\mathbb{L}$, where a configuration is an element of $Ω^Λ$ for some fixed set $Ω$, does there exist a measure $μ$ on configurations on all of $\mathbb{L}$, invariant under some specified symmetry group of $\mathbb{L}$, such that $μ_Λ$ is its marginal on configurations on $Λ$? When the answer is yes, what are the properties, e.g., the entropies, of such measures? Our primary focus is the case in which $\mathbb{L}=\mathbb{Z}^d$ and the symmetries are the translations. For the case in which $Λ$ is an interval in $\mathbb{Z}$ we give a simple necessary and sufficient condition, local translation invariance (LTI), for extendibility. For LTI measures we construct extensions having maximal entropy, which we show are Gibbs measures; this construction extends to the case in which $\mathbb{L}$ is the Bethe lattice. On $\mathbb{Z}$ we also consider extensions supported on periodic configurations, which are analyzed using de~Bruijn graphs and which include the extensions with minimal entropy. When $Λ\subset\mathbb{Z}$ is not an interval, or when $Λ\subset\mathbb{Z}^d$ with $d>1$, the LTI condition is necessary but not sufficient for extendibility. For $\mathbb{Z}^d$ with $d>1$, extendibility is in some sense undecidable.

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