Solvability of Some Statistical Mechanical Systems

We describe a numerical procedure that clearly indicates whether or not a given statistical mechanical system is solvable (in the sense of being expressible in terms of $D$-finite functions). If the system is not solvable in this sense, any solution that exists must be expressible in terms of functions that possess a natural boundary. We provide compelling evidence that the susceptibility of the two-dimensional Ising model, the generating function of square lattice self-avoiding walks and polygons and of hexagonal lattice polygons, and directed animals are in the “unsolvable” class.