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.

28 pages, LaTex, 3 files; significant amount of new material added

References in corpus (1)

• Realizability of point processes

Cited by in corpus (3)

• Entanglement and Nonlocality in Infinite 1D Systems
• Entanglement marginal problems
• Two Dimensional Translation-Invariant Probability Distributions: Approximations, Characterizations and No-Go Theorems