Abstract. We study the Ising model two-point diagonal correlation function C(N, N ) by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable λ, the j-particle contributions, f N,N is expressed polynomially in terms of the complete elliptic integrals E and K. The scaling limit of these differential operators breaks the direct sum structure but not the "Russian doll" structure. The previous λ-extensions, C(N, N ; λ) are, for singledout values λ = cos(πm/n) (m, n integers), also solutions of linear differential equations. These solutions of Painlevé VI are actually algebraic functions, being associated with modular curves.

We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N 6 ) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high-and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant 1 scaling fields, arising first at order |T − T c | 9/4 , though high-low temperature symmetry is still preserved. At terms of order |T − T c | 17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T − T c ) p (log |T − T c |) q with p ≥ q 2 . Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.

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We report computations of the short-and long-distance (scaling) contributions to the square-lattice Ising susceptibility. Both computations rely on summation of correlation functions, obtained using nonlinear partial difference equations. In terms of a temperature variable t, linear in T ͞T c 2 1, the short-distance terms have the form t p ͑lnjtj͒ q with p $ q 2 . A high-and low-temperature series of N 323 terms, generated using an algorithm of complexity O͑N 6 ͒, are analyzed to obtain the scaling part, which when divided by the leading jtj 27͞4 singularity contains only integer powers of t. Contributions of distinct irrelevant variables are identified and quantified at leading orders jtj 9͞4 and jtj 17͞4 .