Convergence of the sum-product algorithm

We address the question of convergence in the sum-product algorithm. Specifically, we relate convergence of the sum-product algorithm to the existence of a weak limit for a sequence of Gibbs measures defined on the associated computation tree. Using tools from the theory of Gibbs measures we develop easily testable sufficient conditions for convergence. The failure of convergence of the sum-product algorithm implies the existence of multiple phases for the associated Gibbs specification. These results give new insight into the mechanics of the algorithm.