Translation Invariant Extensions of Finite Volume Measures

S. Goldstein, T. Kuna, J. L. Lebowitz, E. R. Speer

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6 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)765-782
Number of pages18
JournalJournal of Statistical Physics
Issue number3-4
StatePublished - Feb 1 2017

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Bethe lattice
  • Local translation invariance
  • Maximal entropy extensions
  • Translation invariant extensions
  • de Bruijn graphs


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