Translation invariant extensions of finite volume measures

TL;DR

This paper studies when local measures on lattice subsets can be extended to global translation-invariant measures, characterizing conditions, entropy properties, and the complexity of such extensions in various lattice settings.

Contribution

It introduces the local translation invariance (LTI) condition for intervals in , constructs maximal entropy Gibbs extensions, and analyzes the complexity of extendibility in higher dimensions.

Findings

LTI is necessary and sufficient for  intervals

Maximal entropy extensions are Gibbs measures

Extendibility is undecidable in higher dimensions

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.