0 5 10 15 0 0.5 1 T < | m | > 0 5 10 15 0 10 20 30 T ˜ χ 7 8 9 10 11 12 0 0.1 0.2 0.3 0.4 0.5 T U L 8 FIG. 6: (Color online) Binder cumulant U L as a function of temperature for a d$_{0}$ = 1 system with J$_{0}$ < 0. The average connectivity is c = 1 . 4 and J$_{0}$ = − 0 . 5, J/J$_{0}$ = − 20. The ver- tical line is the theoretical prediction for the critical tempera- ture and the data points correspond to different system sizes, L = 512( . ) , 1024( ◦ ) , 2048(+) , 4096( × ) , 8192( ⋄ ) , 16384( △ ). FIG. 5: (Color online) Magnetization (top) and susceptibility (bottom) as a function of temperature for a d$_{0}$ = 1 system with J$_{0}$ < 0. The average connectivity is c = 1 . 4 and J$_{0}$ = − 0 . 5, J/J$_{0}$ = − 20. The line is the theoretical prediction and the data points correspond to different system sizes, L = 512( . ) , 1024( ◦ ) , 2048(+) , 4096( × ) , 8192( ⋄ ) , 16384( △ ). 0 . 23 , 0 . 21 , 0 . 23 , 0 . 21. These values are smaller than the values in the range 0 . 27 − 0 . 3 that we measured in sec- tion IV A for J$_{0}$ > 0. In the Fig. 6 we plot the Binder cumulant as a function of temperature for the system sizes studied. J/J$_{0}$ = − 20 and c = 1 . 4. We made averages over dis- order by considering 50 samples and we studied system sizes L = 512 , 1024 , 2048 , 4096 , 8192 and 16384. In Fig, 5 we plot the magnetization and the susceptibility as a function of temperature, T . The two critical points are predicted to occur at T$_{c,}$$_{1}$ = 2 . 985 and T$_{c,}$$_{2}$ = 9 . 207. The predicted values of the magnetization in the intermediate temperature range, T$_{c,}$$_{1}$ < T < T$_{c,}$$_{2}$ , are different from the simulation results (see Fig. 5). However, the theo- retical predicted susceptibility, in both P phases, is very close to the simulation susceptibility approaching each other as the system size increases. Note that this model is a frustrated system so that large fluctuations and strong finite size effects are present, especially close to the lower temperature critical point where we do not perform high precision simulation as it requires averaging over a large number of samples. We have studied, in detail, the high temperature criti- cal point where we applied the cumulant crossing tech- nique. The intersection temperatures of the cumulants for L = 512 , 1024 , 2048 , 4096 with the system of size L = 16384 were, 9 . 24 , 9 . 35 , 9 . 23 , 9 . 32, respectively, from which we can estimate a critical temperature equal to 9 . 29(6). This estimate is close but slightly higher than the predicted critical temperature, T$_{c,}$$_{2}$ = 9 . 207. How- ever, considering the statistical error and the finite size corrections we cannot exclude a convergence toward the theoretical value in the thermodynamic limit. The cor- responding intersection values of the cumulant were, We considered the case for c = 10, J$_{0}$ = − 0 . 9 and J = 0 . 5 at d$_{0}$ = 1. From the theory it turns out that for temperatures above 2 . 38 only the zero magnetization solution is stable but for temperatures lower than this value there are always two stable solutions, one with non- zero magnetization and another with zero magnetization. For temperatures T < 2 . 34 the nonzero magnetization solution has the lower free-energy so that a first-order phase transition is predicted at this temperature. The theory also predicts a possible spin-glass phase transition at a temperature, T$_{c,SG}$ = 1 . 88. Note that the theory always predicts continuous spin-glass phase transitions. We made simulations for systems of size L = 512, 1024, 2048, 4096 and 8192 and as before we neglected 10 5 MCS/N for equilibration purposes and we made mea- surements for 10 6 MCS/N. The averaging over disorder was made by considering 50 samples. The results show a first order phase transition occurring at slightly lower temperatures than the one predicted by the theory. The probability distribution of the magnetization clearly ex- hibits (see Fig. 7) the behavior characteristic of first- order phase transitions [30] namely the emergence, at temperatures close to the transition, of two maximum lo- cated at symmetric nonzero values together with a third E. Negative short-range coupling, first-order and spin-glass phase transitions