Some formulas are given for the asymptotic form of N ×N modified block Toeplitz determinants for large N. These formulas are useful in obtaining the asymptotic behaviors of the correlation function of two spins of the Ising model on a two-dimensional regular lattice when the two spins are not on an axis.
We study the asymptotic behavior, for large separations, of the spin-spin correlation function $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ in the two-dimensional Ising model, where the two spins are not necessarily on the same row. Besides the limiting value for infinite separation, which is the square of the spontaneous magnetization, we evaluate the two leading terms in the asymptotic expression in each of the two cases T<Tc and T>Tc. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case T>Tc, but much more complicated for T<Tc. In an Appendix, we also give exactly in a very simple form the correlation $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{N,N}〉$ for symmetrical Ising lattice at the critical temperature Tc.
We study in detail the asymptotic behavior, for large separations, of the correlation between two spins in the same row of the two-dimensional Ising model with the particular value of a purely imaginary magnetic field $H=\frac{1}{2}i\ensuremath{\pi}\mathrm{kT}$. We derive the partition function and the magnetization per spin first proposed by Lee and Yang and find that the correlation function approaches its limiting value exponentially. In contrast to the case of no magnetic field, the approach is shown to be oscillatory rather than monotonic. The first three terms of the asymptotic behavior are explicitly given. The correlation function is obtained by expressing it as a block Toeplitz determinant whose behavior for large separation is studied by solving a pair of coupled Wiener-Hopf sum equations.
We study the boundary correlation function of the fixed-to-free boundary-condition-changing (bcc) operators in the square-lattice Ising model. First, we find a formula for representing a large class of two-point boundary correlation functions using a 2 × 2 block Toeplitz determinant. Using this formula the correlation function of the fixed-to-free bcc operator is represented using block Toeplitz determinants, for arbitrary, uniformly anisotropic couplings. This block Toeplitz determinant is transformed into a scalar Toeplitz determinant when the size of the matrix is an even number. We use Szegö’s theorem and the Fisher–Hartwig theorem to identify the asymptotic behavior of this scalar Toeplitz determinant.
We study in detail the asymptotic behavior, for large separations, of the correlation between two spins in the case of the two-dimensional Ising model without magnetic field. In the limit of infinite separation, this correlation is equal to the square of the spontaneous magnetization per spin. This paper is devoted to answering the question how, for fixed temperature, the correlation approaches this limiting value. The investigation is restricted to the situation where the two spins under consideration lie on the same row of the two-dimensional lattice. There are three distinct cases where the fixed temperature is above, below, or equal to the critical temperature or the Curie temperature. In the first two cases, the limiting value is approached exponentially as a function of the separation, while at the critical temperature the correlation behaves asymptotically as the inverse fourth root of the separation. In this paper, we evaluate exactly the first four terms of the asymptotic expansion in the first case, the first three terms in the second case, and the coefficients of the terms proportional to the -\textonequarter{} and the $\ensuremath{-}\frac{9}{4}$ powers of the separation in the third case. All these results are obtained by first expressing the correlation as a Toeplitz determinant, and the method used is capable of giving in principle the entire asymptotic expansion for large classes of such determinants, when the size of the determinant approaches infinity. The explicit results for the two-dimensional Ising model also serve as an example where the prescription of summing the leading terms, or the most divergent terms, fails.