Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model. V

In a previous paper of this series, we studied $\mathfrak{P}\left(\overline{\sigma }\right)$, the probability that the average boundary spin for a half-plane of Ising spins is $\overline{\sigma }$. We extend our study of such functions by generalizing the previous derivation to any magnetic system and using them to relate the magnetic behavior at the critical isotherm to the spin-spin correlations at the critical temperature. We also clarify the meaning of the secondary maxima previously found in $\mathfrak{P}\left(\overline{\sigma }\right)$ by a more accurate calculation and by examining the magnetization on interior rows of the half-plane. These considerations show that $\mathfrak{P}\left(\overline{\sigma }\right)$ is more properly to be considered as the sum of two separate spin probability functions. The various ways in which the thermodynamic limit may be taken and the effects that these different limiting procedures have on $\mathfrak{P}\left(\overline{\sigma }\right)$ are discussed in detail. The influence of boundary conditions on spin probability functions is studied by reversing the sign of one column of horizontal bonds. We compute the additional free energy resulting from the misfit bonds and use this to show that $\mathfrak{P}\left(\overline{\sigma }\right)$ is no longer bimodal but is rectangular for $\left|\overline{\sigma }\right|$ less than the boundary spontaneous magnetization. Finally, we present in graphical form numerical integrations of the boundary magnetization and susceptibility.