Ferromagnetic ordering in graphs with arbitrary degree distribution

Abstract

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 minimum and maximum degree, and the degree distribution characterizing the graph. As expected, there is a ferromagnetic transition provided <k> <= <k^2> < \infty. However, if the fourth moment of the degree distribution is not finite then non-trivial scaling exponents are obtained. These results are analyzed for the particular case of power-law distributed random graphs.