MAGNETIZATION IN THE ZIG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS

— We discuss the magnetization Mm in the m-th column of the zig-zag layered 2D Ising model on a half-plane using Kadanoff–Ceva fermions and orthogonal polynomials techniques. Our main result gives an explicit representation of Mm via m × m Hankel determinants constructed from the spectral measure of a certain Jacobi matrix which encodes the interaction parameters between the columns. We also illustrate our approach by giving short proofs of the classical Kaufman–Onsager–Yang and McCoy–Wu theorems in the homogeneous setup and expressing Mm as a Toeplitz+Hankel determinant for the homogeneous sub-critical model in presence of a boundary magnetic field.