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.

Figure 1

Finite-size corrections of the ground-state energy density in the $T=0$ RFIM on Erdös-Rényi random graphs with mean connectivity $z=4$. (top) Numerical data for different system sizes and the analytical formula given by Eq. (42). Close to the critical point (which is $J_c\approx 0.395$) the scaling of the energy corrections is given by the mean-field prediction (45), as confirmed by the data collapse shown in the inset. (bottom) Estimates of formula (42), truncating the sum over loops with a cutoff $L=3,4,5,6,7$ and extrapolating the whole series as explained in the main text.

Figure 2

Binder cumulant for the $T=0$ RFIM on Erdös-Rényi random graphs with mean connectivity $z=4$ for different system sizes as a function of the exchange interaction $J$. A vertical dashed line is drawn in correspondence to the critical point $J_c\sim 0.395$ . The inset shows the data collapse in the critical region using the scaling variable $(J-J_c)N^{1/3}$ for the reduced interaction.

Figure 3

Average squared magnetization for the $T=0$ RFIM on Erdös-Rényi random graphs with mean connectivity $z=4$ for different system sizes as a function of the exchange interaction $J$. The inset shows the data collapse in the critical region using the scaling $N^{\frac{1}{3}}$ for both the reduced interaction and the squared magnetization.