6 TABLE I: (Color online) Results of Binder cumulant cross- ings for d$_{0}$ = 2 and J$_{0}$ > 0. In the column β$_{L,L}$ ′ J$_{0}$ we list the crossing inverse temperatures between the cumulants for systems of side L with L ′ = 128. In the column β c J$_{0}$ we list the critical parameters obtained from Eq. (27). The crossing values of the cumulant are listed in the column U ∗ L,L ′ . FIG. 2: (Color online) In (a) we plot the susceptibility, ˜ χ c , and in (b) the magnetization, 〈| m |〉$_{c}$ , at the critical point for the d$_{0}$ = 2 model with J/J$_{0}$ = 1 as a function of the system side L . Both in (a) and (b) the data are for c = 0( ⋆ ) , 0 . 5( × ) , 1( ○ ) , 1 . 5(+) , 2( □ ). 2 2.5 3 3.5 4 4.5 5 0 5 10 (a) ln L ln ˜ χ c 2 2.5 3 3.5 4 4.5 5 −3 −2 −1 0 (b) ln L ln < | m | > c c L β$_{L,L}$ ′ J$_{0}$ β c J$_{0}$ U ∗ L, 128 16 0.4410 0.612 0 32 0.4409 0 . 440686 ... 0.612 64 0.4411 0.615 16 0.2968 0.33 0.5 32 0.2964 0.2963 0.30 64 0.2960 0.27 16 0.2466 0.32 1.0 32 0.2461 0.2461 0.28 64 0.2460 0.28 16 0.2139 0.30 1.5 32 0.2137 0.2134 0.29 64 0.2136 0.28 16 0.1894 0.30 2.0 32 0.1891 0.1897 0.27 64 0.1888 0.24 16 0.1173 0.28 5.0 32 0.1174 0.1167 0.29 64 0.1173 0.27 16 0.0730 0.28 10.0 32 0.0730 0.0728 0.28 64 0.0731 0.30 the curves C $^{F}$( r ), for c > 0, approach the data points for c = 0 calculated for a system of size L = 64. Note that, as we approach the critical temperature, the correlation function at large r reaches a finite value that decreases as the system size increases. This finite constant is just due to the finite size of the system and it is predicted by the theory to vanish as 1 /N , in the thermodynamic limit, but at the same time it is responsible for the divergence, with system size, of the susceptibility at the critical point (see Eq. (33) and subsequent comments of Ref. [12]). expected value − 0 . 5; whereas for the susceptibility we ob- tain the exponents 1 . 03(5) , 1 . 06(7) , 0 . 97(2) and 1 . 01(5), also close to the expected result for mean-field behavior. From Eq. (18) we see that the effective field theory, in the P region and zero external field, predicts the spin- spin correlation function of the random model C ( F ) to be, in the thermodynamic limit, equal to the correlation function of the pure model calculated at the same tem- perature. In order to check this result we calculated, from simulation, In the P phase at zero external magnetic field the effec- tive field theory prediction for the susceptibility is exact, consistently with the exact predictions for the critical temperatures. The question remains whether the sus- ceptibility prediction is a good approximation for non- zero field. To verify this we made simulations for the case J$_{0}$ = 0 with a positive long-range coupling (as in Eq. (10)) (in other words the simplest version of Viana- Bray model). For this particular case the magnetization is predicted to be given by the solution of the following equation where σ$_{0}$ is an arbitrary spin and σ$_{r}$ is one spin at Euclidean distance r from the spin σ$_{0}$ , measured on the lattice L$_{0}$ . We considered the case d$_{0}$ = 2 with J = J$_{0}$ and c = 1 and 2 and the inverse temperatures βJ$_{0}$ = 0 . 1 , 0 . 17571. These temperatures are above the critical temperatures for the two values of c studied. We studied also the correlation function at the critical in- verse temperatures β$_{c}$J$_{0}$ = 0 . 2461 and β$_{c}$J$_{0}$ = 0 . 1897 for c = 1 and c = 2, respectively (see Table I). These calcu- lations were done for system sides L = 8 , 16 , 32 and 64. In Fig. 3 we can see that, as the system size increases, and the susceptibility (divided by β ) is given by, ˜ χ ( βJ, βh ) = 1 − m 2 1 − c (1 − m $^{2}$) tanh βJ (33) m = tanh ( c m tanh( βJ ) + βh ) (32) C ( F $^{)}$( r ) = 〈 σ$_{0}$σ$_{r}$ 〉 , (31) C. Susceptibility and Magnetization at non-zero external field above the critical temperature B. Correlation functions above the critical temperature