  Loop percolation, also known as the dense O(1) loop model, is a variant of
critical bond percolation in the square lattice 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.
