Finite size effects for the Ising model on random graphs with varying dilution

We investigate the finite size corrections to the equilibrium magnetization of an Ising model on a random graph with N nodes and N gamma edges, with 1 < gamma <= 2. By conveniently rescaling the coupling constant, the free energy is made extensive. As expected, the system displays a phase transition of the mean-field type for all the considered values of gamma at the transition temperature of the fully connected Curie-Weiss model. Finite size corrections are investigated for different values of the parameter gamma, using two different approaches: a replica based finite N expansion, and a cavity method. Numerical simulations are compared with theoretical predictions. The cavity based analysis is shown to agree better with numerics. (C) 2009 Elsevier B.V. All rights reserved