Boundary magnetization and spin correlations in inhomogeneous two-dimensional Ising systems

We consider a system of Ising spins on a semi-infinite triangular lattice with nearest-neighbor coupling constants that depend on the distance $m$ from the boundary. An exact technique for calculating the boundary magnetization and the boundary pair correlation function is described. It relies on repeated application of a mapping based on the star-triangle transformation. The case of coupling constants that differ from the bulk coupling by an amount ${\mathrm{Am}}^{-y}$ for large $m$ is examined in detail. For $y<\mathrm{}1\mathrm{}$ the inhomogeneity of the couplings leads to an interesting variety of modifications in the boundary critical behavior. For $y=1\mathrm{}$, $A>\mathrm{}{A}_c$ ($A_c$ being a positive critical value), and for $y<\mathrm{}1\mathrm{}$, $A>\mathrm{}0\mathrm{}$, there is a spontaneous surface magnetization at the bulk critical temperature.