The Belief Propagation algorithm is an iterative procedure on R2n which aims to approximate the Gibbs state of an Ising model on a graph with n edges. Key ideas and the unusual name came from the work of a mathematical philosopher Judea Pearl. The algorithm is of considerable significance for applications in statistical inference, combinatorial optimization and image processing. It is known to converge to a fixed point unless it does not and presents a challenge for a dynamicist. The deepest insight into its dynamics so far comes in the form of a variational principle, namely that fixed points of the algorithm are stationary points of a certain functional called Bethe's free energy.

In the talk, beyond providing the necessary background and definitions, I will discuss certain variants of the algorithm for which, unlike in the classical formulation, Bethe's free energy changes monotonically under the dynamics. Based on this approach one finds fixed points of BP with non-constant spontaneous magnetization in the absence of an external field.