7 0 0.5 1 0 0.5 1 (  J) −1 =1.2903 (a)  h m 0 0.5 1 0 1 2 3 (d) (  J) −1 =1.2903  h ˜ χ 0 0.5 1 0 0.5 1 (  J) −1 =1.8182 (b)  h m 0 0.5 1 0 5 10 (e) (  J) −1 =1.8182  h ˜ χ 0 0.5 1 0 0.5 1 (c) (  J) −1 =2.1739  h m 0 0.5 1 0 5 10 (f) (  J) −1 =2.1739  h ˜ χ 0 2 4 5 10 −4 10 −2 10 0 (a) βJ 0 =0.1 r C (F) (r) 0 2 4 5 10 −4 10 −2 10 0 (b) βJ 0 =0.17571 r C (F) (r) 0 2 4 5 10 −4 10 −2 10 0 (c) βJ 0 =0.2461 r C (F) (r) 0 2 4 5 10 −4 10 −2 10 0 (d) βJ 0 =0.1 r C (F) (r) 0 2 4 5 10 −4 10 −2 10 0 (e) βJ 0 =0.17571 r C (F) (r) 0 2 4 5 10 −4 10 −2 10 0 (f) βJ 0 =0.1897 r C (F) (r) FIG. 4: (Color online) Magnetization ( (a), (b) and (c) ) and Susceptibility ( (d), (e), (f)) for the Viana-Bray model and c = 2 for three temperatures as a function of the ex- ternal magnetic field. The lines are the theoretical predic- tions ( see Eqs. (32) and (33)) and the data points are sim- ulation results for N = 128( . ) , 256( ◦ ) , 512(+) , 1024( × ) and 2048( ♦ ). For the case studied here the critical temperature is, ( β c J ) − 1 = 1 . 8205. FIG. 3: (Color online) Correlation functions C ( F $^{)}$( r ) as a function of Euclidean distance r for J = J$_{0}$ and c = 1 (in (a), (b) and (c)) and c = 2 (in (d), (e) and (f)) at different temperatures, βJ$_{0}$ = 0 . 1 ((a) and (d)), 0 . 17571 ( (b) and (e)). In (c) with β c J$_{0}$ = 0 . 2461 and in (f) with β c J$_{0}$ = 0 . 1897 (the critical temperatures for c = 1 and c = 2, respectively). The lower line (which is blue in the online version of this paper) is C ( F $^{)}$( r ) for the pure model c = 0 and for a system side L = 64 ( ♦ ). Data for four system sides are plotted, L = 8 ( ○ ), L = 16 (+), L = 32 ( × ), L = 64 ( ♦ ). It is worth to observe that with respect to our effective field theory, the Viana-Bray model represents the worst, i.e. , the most difficult, case. The theory in fact, by con- struction, takes exactly into account all the effects due to the short-range couplings and to the short-loops present in the given lattice L$_{0}$ , and the greater is d$_{0}$ , the greater is the level of accuracy of the theory also out of the P region (at least in the absence of frustration), while in the Viana-Bray model topologically we have d$_{0}$ = 0. In Fig. 4 we compare the results of the above predic- tions for the magnetization and susceptibility with sim- ulation results for c = 2. For averaging purposes we con- sidered 10 samples. The plots correspond to three tem- peratures ( βJ ) − 1 = 1 . 2903 (first row), ( βJ ) − 1 = 1 . 8182 (middle) and ( βJ ) − 1 = 2 . 1739 (bottom). The critical temperature is ( β$_{c}$J ) − 1 = 1 . 8205. By construction, in the limit of strong field the magnetization prediction be- comes exact and similarly in the limit of small field above the critical temperature. In the intermediate field range we see that the magnetization prediction and simulation results in general do not agree. However, above the criti- cal temperature, the susceptibility obtained from simula- tion and the theoretical prediction given by Eq. (33) are very close to each other over the full range of field val- ues studied. Note that for ( βJ ) − 1 = 1 . 8182, close to the critical temperature, the simulation susceptibility shows, as expected, a strong finite-size effect at zero field. In the case J$_{0}$ < 0 the theory allows for the occur- rence of two second-order phase transitions. At low and high temperatures the system is disordered and in the intermediate temperatures a ferromagnetic phase arises. The simulations confirm this phase diagram pic- ture. We made simulations at d$_{0}$ = 1, for J$_{0}$ = − 0 . 5, D. Negative short-range coupling and second order phase transitions