Exact and approximate solutions for the dilute Ising model

The ground state energy and entropy of the dilute mean field Ising model is computed exactly by a single order parameter as a function of the dilution coefficient. An analogous exact solution is obtained in the presence of a magnetic field with random locations. Results lead to a complete understanding of the geography of the associated random graph. In particular, we give the size of the giant component (continent) and the number of isolated clusters of connected spins of all given size (islands). We also compute the average number of bonds per spin in the continent and in the islands. Then, we tackle the problem of solving the dilute Ising model at strictly positive temperature. In order to obtain the free energy as a function of the dilution coefficient and the temperature, it is necessary to introduce a second order parameter. We are able to find out the exact solution in the paramagnetic region and exactly determine the phase transition line. In the ferromagnetic region we provide a solution in terms of an expansion with respect to the second parameter which can be made as accurate as necessary. All results are reached in the replica frame by a strategy which is not based on multi-overlaps.