Ferromagnetic ordering in graphs with arbitrary degree distribution

We present a detailed study of the phase diagram of the Ising model in random graphs with arbitrary degree distribution. By using the replica method we compute exactly the value of the critical temperature and the associated critical exponents as a function of the moments of the degree distribution. Two regimes of the degree distribution are of particular interest. In the case of a divergent second moment, the system is ferromagnetic at all temperatures. In the case of a finite second moment and a divergent fourth moment, there is a ferromagnetic transition characterized by non-trivial critical exponents. Finally, if the fourth moment is finite we recover the mean field exponents. These results are analyzed in detail for power-law distributed random graphs.