Translation Invariant Extensions of Finite Volume Measures

We investigate the following questions: Given a measure μΛ on configurations on a subset Λ of a lattice 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 L, invariant under some specified symmetry group of L, such that μΛ 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 L= Zd and the symmetries are the translations. For the case in which Λ is an interval in 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 L is the Bethe lattice. On 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 ⊂ Z is not an interval, or when Λ ⊂ Zd with d> 1 , the LTI condition is necessary but not sufficient for extendibility. For Zd with d> 1 , extendibility is in some sense undecidable.