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Generalisation of the Hammersley-Clifford theorem on bipartite graphs


Author: Nishant Chandgotia
Journal: Trans. Amer. Math. Soc. 369 (2017), 7107-7137
MSC (2010): Primary 60K35; Secondary 82B20, 37B10
DOI: https://doi.org/10.1090/tran/6899
Published electronically: May 1, 2017
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Abstract: The Hammersley-Clifford theorem states that if the support of a Markov random field has a safe symbol, then it is a Gibbs state with some nearest neighbour interaction. In this paper we generalise the theorem with an added condition that the underlying graph is bipartite. Taking inspiration from Brightwell and Winkler (J. Combin. Theory Ser. B 78 (2000), 141-166) we introduce a notion of folding for configuration spaces called strong config-folding proving that if all Markov random fields supported on $ X$ are Gibbs with some nearest neighbour interaction, then so are Markov random fields supported on the `strong config-folds' and `strong config-unfolds' of $ X$.


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Additional Information

Nishant Chandgotia
Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Email: nishant.chandgotia@gmail.com

DOI: https://doi.org/10.1090/tran/6899
Keywords: Markov random fields, Gibbs states, nearest neighbour interaction, folding, dismantlable, Hammersley-Clifford, symbolic dynamics
Received by editor(s): June 6, 2014
Received by editor(s) in revised form: November 30, 2015
Published electronically: May 1, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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