Date & Time: Monday, August 10, 2015, 16:30-17:30.
Venue: Ramanujan Hall
Title: The Hammersley-Clifford Theorem
Speaker: Nishant Chandgotia, University of British Columbia.
Abstract: A Markov random field is a probability measure supported on a space of configurations on G satisfying a conditional independence property: given two finite separated sets A and B contained in G the configuration on A and B are independent conditioned on the configuration on their complement. If the underlying support has a “safe symbol” it is known via the Hammersley-Clifford theorem that the measure is actually a Gibbs state. We want to identify spaces of configurations which satisfy the conclusion of this theorem, that every Markov random field supported on it is a Gibbs state for some nearest neighbour interaction. After a brief overview of the area, we will discuss two such conditions: 1) When G is bipartite we provide a generalisation of the Hammersley-Clifford theorem 2) When G=Z^d we prove that the space of homomorphisms from G to an n-cycle satisfies the conclusion of the theorem but lies beyond the aforementioned generalisation.