0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 c c  J 0 5 we analyze the spin-spin correlation function above the critical temperature for d$_{0}$ = 2. In subsection IV C for the special case J$_{0}$ = 0 we study the magnetization and susceptibility at non-zero external field above the criti- cal temperature. In subsection IV D we consider a one dimensional system with J$_{0}$ negative where two second- order phase transitions are predicted. In subsection IV E we analyze the same one-dimensional model for couplings and connectivity such that either two first-order phase transitions or a spin-glass phase are predicted. Finally in subsection IV F we study a two dimensional Edwards- Anderson model with added long-range shortcuts. From the susceptibility of the pure Ising model in a hypercubic lattice of dimensionality d$_{0}$ and Eq. (27) we obtain the location of the P-F line of critical points in the c-T plane. For d$_{0}$ = 1 we use the known analytical expression for the susceptibility, ˜ χ$_{0}$ ( βJ$_{0}$, 0) = exp(2 βJ$_{0}$ ), that applied to Eq. (27) reproduces the same formula of [9] for the P-F and P-SG lines. For higher dimensions we use numerical results obtained from Metropolis Monte- Carlo simulations. For d$_{0}$ = 2 the pure model susceptibility was de- termined for several systems sides up to L = 128 to check for finite-size effects. The pure model was sim- ulated for 40 temperatures, in the range 0 . 1 < βJ$_{0}$ < = β$_{c}$$_{0}$J$_{0}$ = 0 . 44068 ... . For d$_{0}$ = 3 we studied systems of side L = 8 and L = 16 also for 40 temperatures in the range 0 . 05 < βJ$_{0}$ < = β$_{c}$$_{0}$J$_{0}$ = 0 . 2216546(10) [26]. In all of the simulations reported we neglected the first 10 5 MCS/N and made measurements in the remaining 10 6 MCS/N steps. In Fig. 1 we plot the critical lines for d$_{0}$ = 1 with J/J$_{0}$ = 3 / 5, and for d$_{0}$ = 2 and 3 with J/J$_{0}$ = 1. Note that in the figure, for d$_{0}$ = 2 and 3, there are several lines corresponding to the use of susceptibility estimates ob- tained from systems of different size. Nevertheless, in the scale of the plot no finite size effects can be seen. This is a consequence of the fact that only very near c = 0 the solution of Eq. (27) uses values of the pure model suscep- tibility near the critical point. Far from the critical point of the pure model the susceptibility and consequently the estimates of the critical line in the disordered model do not show finite-size effects. In order to compare the predictions based on Eq. (27) we measured by direct simulation the location of the crit- ical points for several values of the average connectivity, c = 0 , 0 . 5 , 1 , 1 . 5 , 2 , 5 and 10 and the results, for each spa- tial dimension, are plotted in Fig. 1. The average over disorder was done by considering averages over 10 sam- ples. For the case d$_{0}$ = 2 we present in Table I detailed nu- merical results. For the cumulant crossing β$_{L,L}$ ′ J$_{0}$ (third column) listed for several values of c we estimate a statis- tical error 0 . 0005 which allows us to claim a good agree- Furthermore, for d$_{0}$ = 2, we measured the scaling with system size of the average value of the absolute value of the magnetization 〈| m |〉 ( β$_{c}$J$_{0}$ ) and the susceptibility, ˜ χ ( β$_{c}$J$_{0}$ ) at the critical point. For a mean-field universal- ity class these quantities are expected to scale at critical- ity like 〈| m |〉$_{c}$ ∼ N − 1 / 4 ∼ L − d$_{0}$/ 4 and ˜ χ$_{c}$ ∼ N 1 / 2 ∼ L d$_{0}$/ 2 where N is the total number of spins [28, 29, 31]. In Fig. 2 we show these quantities plotted in bi-logarithmic scale as a function of system size for the different c values studied. For the magnetization, the slope of the straight line fit for c = 0 is − 0 . 125(1) and for the susceptibility is 1 . 752(1), consistent with the known exact exponents of the pure d$_{0}$ =2 Ising model. For c = 0 . 5 , 1 , 1 . 5 and 2 we got, respectively, for the magnetization exponents − 0 . 49(3) , − 0 . 48(3) , − 0 . 52(2) and − 0 . 48(3) close to the The mean-field value of the Binder cumulant at criti- cality for an infinite system is predicted to be 0 . 2705[28, 29, 31] which is close to our estimates of Binder cumulant intersections listed in table I (fifth column) for which we estimate an error equal to 0 . 02. In three dimensions we simulated only systems of side L = 8 and 16. For c = 0 the two cumulants intersect at the value U$_{L}$ = 0 . 486 close to the estimation 0 . 46521 reported in [26]. For the other values of c studied, c = 0 . 5 , 1 , 1 . 5 , 2 , 5 and 10 the inter- section was measured near U$_{L}$ = 0 . 31. In one dimension we studied only c = 0 . 5 , 5 and c = 10 and also the Binder cumulant intersections were found to be near 0 . 3 for in- tersections of L = 128 , 256 , 512 , 1024 with L ′ = 2048. ment between the simulations and the theoretical predic- tion obtained from Eq. (27) (fourth column). The values of the cumulant at the critical point for the pure model ( c = 0) are close to the value 0 . 61069 calculated in [27]. The long-range links introduced by the disorder change the universality class from 2d Ising to mean-field. FIG. 1: (Color online) Lines of critical points in the β − c plane for d$_{0}$ = 1 ( ○ ) with J/J$_{0}$ = 3 / 5, and for d$_{0}$ = 2 ( □ ) and d$_{0}$ = 3 ( ⋆ ) with J/J$_{0}$ = 1 as obtained from the pure model susceptibility and Eq. (27). The symbols are estimates of the critical points obtained from simulation and by using the Binder cumulant intersection technique. A. The Paramagnetic-Ferromagnetic line of critical points