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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 $N^γ$ edges, with $1 < γ\leq 2$. 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 $γ$ at the transition temperature of the fully connected Curie-Weiss model. Finite size corrections are investigated for different values of the parameter $γ$, 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.