Generalisation of the Hammersley-Clifford theorem on bipartite graphs

Chandgotia, Nishant

doi:10.1090/tran/6899

Generalisation of the Hammersley-Clifford theorem on bipartite graphs
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Trans. Amer. Math. Soc. 369 (2017), 7107-7137 Request permission

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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