Phase diagram of the Ising model on percolation clusters

The annealed Ising magnet on percolation clusters is studied by means of a mapping into a Potts-Ising model and with the Migdal-Kadanoff renormalization-group method. The phase diagram is determined in the three-dimensional parameter space of the Ising coupling K, the bond-occupation probability p, and the fugacity q, which controls the number of clusters. Three phases are identified: percolating ferromagnetic, percolating paramagnetic, and nonpercolating paramagnetic. For large q the phase diagram includes a multicritical point at the intersection of the Ising critical line and the percolation critical line. In the case of random bond percolation (q=1) the Ising critical line is: p(1-${\mathit{e}}^{\mathrm{-}2\mathit{K}}$)=1-exp(-2${\mathit{L}}_{\mathit{C}}$), where ${\mathit{L}}_{\mathit{C}}$ is the pure-Ising-model critical coupling.