Monte Carlo study of the square-lattice annealed Ising model on percolating clusters

Simulations of an Ising $q$-state Potts model which is equivalent to the Ising model on annealed percolation clusters are used to determine the phase diagram of the model in two dimensions. Three topologically different phase diagrams are obtained: (i) for $q=2,$ there are two critical Ising lines meeting at $T=0\mathrm{}$ at the four-state Potts critical point; (ii) for $2<q<~4,$ the Ising critical line meets the $q$-state critical line and a line of first-order transitions at a bicritical point; (iii) for $q>\mathrm{}4,$ the Ising critical line intersects a line of first-order transitions at a critical end point.