Solvable Probabilistic Model for Cycles in Planar Graph

Abstract

We study a probabilistic model for single connected cycles on an undirected planar graph for which the degree of every vertex is restricted to two or three. By using a diagrammatical method to solve a free fermion model, we derive the exact expression of the partition function as well as marginal probabilities. We compare the exact results for the marginals with the approximations obtained with Loopy Belief Propagation allowing us to evaluate the efficacy of the latter.