The Thermodynamic Limit -- The Spontaneous Magnetization and Two-Point Spin Correlation -- Scaling Limits -- The One-Point Green Function -- Scaling Functions as Tau Functions -- Deformation Analysis of Tau Functions.

This book examines in detail the correlations for the two-dimensional Ising model in the infinite volume or thermodynamic limit and the sub- and super-critical continuum scaling limits. Steady progress in recent years has been made in understanding the special mathematical features of certain exactly solvable models in statistical mechanics and quantum field theory, including the scaling limits of the 2-D Ising (lattice) model, and more generally, a class of 2-D quantum fields known as holonomic fields. New results have made it possible to obtain a detailed nonperturbative analysis of the multi-spin correlations. In particular, the book focuses on deformation analysis of the scaling functions of the Ising model. This self-contained work also includes discussions on Pfaffians, elliptic uniformization, the Grassmann calculus for spin representations, Weiner--Hopf factorization, determinant bundles, and monodromy preserving deformations. This work explores the Ising model as a microcosm of the confluence of interesting ideas in mathematics and physics, and will appeal to graduate students, mathematicians, and physicists interested in the mathematics of statistical mechanics and quantum field theory.