Translation Invariant Extensions of Finite Volume Measures

Abstract

We investigate the following questions: Given a measure $\mu _\Lambda$ on configurations on a subset $\Lambda$ of a lattice $\mathbb {L}$, where a configuration is an element of $\Omega ^\Lambda$ for some fixed set $\Omega$, does there exist a measure $\mu$ on configurations on all of $\mathbb {L}$, invariant under some specified symmetry group of $\mathbb {L}$, such that $\mu _\Lambda$ is its marginal on configurations on $\Lambda$? 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 $\Lambda$ 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 $\Lambda \subset \mathbb {Z}$ is not an interval, or when $\Lambda \subset \mathbb {Z}^d$ 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.