Connectivity Patterns in Loop Percolation I: the Rationality Phenomenon and Constant Term Identities

Loop percolation, also known as the dense O(1) loop model, is a variant of critical bond percolation in the square lattice \({\mathbb{Z}^2}\) whose graph structure consists of a disjoint union of cycles. We study its connectivity pattern, which is a random noncrossing matching associated with a loop percolation configuration. These connectivity patterns exhibit a striking rationality property whereby probabilities of naturally-occurring events are dyadic rational numbers or rational functions of a size parameter n, but the reasons for this are not completely understood. We prove the rationality phenomenon in a few cases and prove an explicit formula expressing the probabilities in the “cylindrical geometry” as coefficients in certain multivariate polynomials. This reduces the rationality problem in the general case to that of proving a family of conjectural constant term identities generalizing an identity due to Di Francesco and Zinn-Justin. Our results make use of, and extend, algebraic techniques related to the quantum Knizhnik-Zamolodchikov equation.