Annealed Ising model with site dilution on self similar structures

We consider an Ising model on the triangular Apollonian network (AN), with a thermalized distribution of vacant sites. The statistical problem is
formulated in a grand canonical ensemble, in terms of the temperature $T$ and a chemical potential $\mu$ associated with the concentration of active magnetic sites. We also investigate the analogous model on the diamond hierarchical lattice (DHL). We use a well-known transfer matrix method, with a number of adaptations, to derive a set of (non linear, discrete) maps for the free energy and other auxiliary variables along successive generations of the hierarchical structure. The major changes as compared to the quenched or ordered situations amounts to formulating the statistical problem in terms of an effective Hamiltonian in a grand-canonical ensemble, depending on temperature $T$ and a chemical potential $\mu$, which is associated with the concentration of occupied sites. This is justified by the fact that, in a thermalized system, the orientational variables are treated on the same basis as the positional disorder degrees of freedom. From the numerical iteration of the recursion relations, we obtain various thermodynamic quantities. In the $\mu\rightarrow\infty$ limit, we reproduce the results for the uniform models: in the AN, the system is magnetically ordered at all temperatures, while in the DHL there is a ferromagnetic-paramagnetic transition at a finite value of $T$. Magnetic ordering, however, is shown to disappear for sufficiently large negative values of the chemical potential.