Inversion relations, the ising model and polygons

Abstract

For the two-dimensional Ising model, it is known that the existence of an inversion relation, certain symmetries and analyticity properties are sufficient to implicitly define the famous Onsager solution for the free-energy of the Ising model. For the susceptibility of the Ising model, no such solution is known, though various inversion relations are also known. We describe some attempts to exploit these inversion relations to solve the susceptibility, and we also describe attempts to find analogous results for the polygon generating function for square lattice polygons. From this work, it is clear that none of these problems are algebraic, or, indeed, even D-finite.