Bounded and inhomogeneous Ising models. III. Regularly spaced point defects

We calculate exactly the transition temperature for a rectangular lattice Ising model with various types of point defects (including missing sites or vacancies) regularly distributed through the lattice on an $m\times n$ grid. We prove that for concentration $x=\frac{1}{\mathrm{mn}}$, the transition temperature is shifted to $T_c\mathrm{}\left(x\right)=T_c\mathrm{}\left(0\right)[1-Q_{1}x+Q_2{x}^2\ln x-Q_3{x}^2-Q_4{x}^3{\ln }^{2}x+\dots ]$, where the constants $Q_1$, $Q_2$, $Q_4$, and the function $Q_3\left(\frac{n}{m}\right)$ are explicitly derived. The incremental free energies per isolated defect are also calculated. The various amplitudes obtained obey appropriate scaling relations.