Finite-size corrections to disordered systems on Erdös-Rényi random graphs

We study the finite-size corrections to the free-energy density in disordered spin systems on sparse random graphs, using both replica theory and the cavity method. We derive analytical expressions for the O(1/N) corrections in the replica symmetric phase as a linear combination of the free energies of open and closed chains. We perform a numerical check of the formulas on the random-field Ising model at zero temperature by computing finite-size corrections to the ground-state energy density.