**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.