TY - JOUR

T1 - Translation Invariant Extensions of Finite Volume Measures

AU - Goldstein, S.

AU - Kuna, T.

AU - Lebowitz, J. L.

AU - Speer, E. R.

N1 - Funding Information:
The work J.L.L. was supported in part by NSF Grant DMR 1104500 and AFOSR Grant FA9550-16-1-0037. We thank A. C. D. van Enter and M. Hochman for bringing to our attention previous work on this problem, M. Hochman, M. Saks, S. Thomas, and A. C. D. van Enter for helpful discussions, and D. Avis for making the computer program lrs available to the public and for helpful advice on its use.
Publisher Copyright:
© 2016, Springer Science+Business Media New York.

PY - 2017/2/1

Y1 - 2017/2/1

N2 - 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.

AB - 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.

KW - Bethe lattice

KW - Local translation invariance

KW - Maximal entropy extensions

KW - Translation invariant extensions

KW - de Bruijn graphs

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U2 - 10.1007/s10955-016-1595-8

DO - 10.1007/s10955-016-1595-8

M3 - Article

AN - SCOPUS:84981554473

SN - 0022-4715

VL - 166

SP - 765

EP - 782

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 3-4

ER -