Two - Dimensional king Correlations : Convergence of the Scaling Limit *

In this paper we will establish formulas for the correlations of the two-dimensional Ising model in the absence of a magnetic field and prove the convergence of the scaling limit from above and below the critical temperature. The theoretical developments which lead up to our results begin with Onsager’s calculation of the free energy for this model in a classic 1944 paper [52). Statistical mechanics in the infinite-volume limit is expected to exhibit phase transitions through nonanalytic behavior in thermodynamic quantities; the Onsager formula for the free energy as a function of temperature was the first explicit example of such behavior. In a sequel to Onsager’s paper, Kaufman [34] simplified the analysis by emphasizing the role of the spin representations of the orthogonal group; Kaufman and Onsager [35] subsequently used this idea to study the short-range order. By 1949 Onsager [53] knew the formula for the spontaneous magnetization, and Yang gave an independent derivation of this result in 1952 [74]. In [28] Kac and Ward and later in [32] Kasteleyn pioneered a combinatorial attack on the Ising model. Montroll, Potts, and Ward [49] used this method to give formulas for the correlations as Pfaffians. The size of the Pfaffians in these formulas grows with the separation of the sites in the correlations and the asymtotic behavior at large separation (clustering) is far from evident. To go beyond the spontaneous magnetization in the analysis of the clustering of correlations, corrections to the Szego formula were devised. This problem has a long history, and we mention in connection with the Ising model the fundamental papers by Wu [72] and by Kadanoff (291 in 1966, and by Cheng and Wu in 1967 [ 131, and refer the reader to the book by McCoy and Wu [40] for further details up to 1972. In Fisher [ 181 and Kadanoff [30] a notion of scaling for statistical systems near a critical point was proposed. To understand the scaling limit for the