9 1.5 2 2.5 3 0 0.5 1 (a) T < | m | > 1.5 2 2.5 3 10 0 10 2 (b) T ˜ χ 1.5 2 2.5 3 −0.5 0 0.5 1 (c) T U L −1 −0.5 0 0.5 1 0 5 10 15 P ( m ) m −1 −0.5 0 0.5 1 0 2 4 6 8 m P ( m ) FIG. 8: (Color online) Simulation results for the average magnetization (a), magnetic susceptibility (b) and Binder cu- mulant U L (c) as a function of temperature for the model with c = 10, J$_{0}$ = − 0 . 9 and J = 0 . 5 at d$_{0}$ = 1. In (b) we have used ˜ χ = N< m 2 > at any temperature. The vertical lines are at T c = 2 . 34 and T c,SG = 1 . 88. In (b) the line is the theoretical susceptibility for the zero magne- tization solution. The simulation data points are for L = 512( ○ ) , 1024( × ) , 2048( . ) , 4096(+) , 8192( ⋄ ). maximum near zero magnetization. In Fig. 8(a) we show the simulation results for the av- erage magnetization, the magnetic susceptibility (b), and the magnetization Binder cumulant (c). The simulation susceptibility follows very closely the zero magnetization theoretical susceptibility up to temperatures lower than the predicted T$_{c}$ . Note that, in the case under study, the zero magnetization solution is a stable solution at any temperature and the theoretical prediction of the phase transition is based on comparison of the value of free-energies of the solutions. The theory does not pre- dict correctly the location of the transition since values of the free-energy of the non-zero magnetization solu- tion are not given exactly by the theory. The simulation Binder cumulant shows, near the transition temperature, the expected increasingly negative values, as the system size increases [30]. Interestingly, the simulation data give a transition temperature close to the theoretically pre- dicted spin-glass transition temperature. We also studied the overlap order parameter distribu- tion and the results are shown in Fig. 9. Here, also, the behavior expected for a first-order transition temperature was observed. The negative value minima of the Binder cumulant for the overlap order parameter are steeper and occur at slightly higher temperatures, for the same sys- tem sizes, as compared with the corresponding quantity for the magnetization. Quite interestingly for c = 4 . 5 and the values of the coupling constants, J$_{0}$ = − 0 . 9 and J = 1, the theory predicts the absence of ferromagnetic phase transitions and only a continuous P-SG phase transition located at T ( SG ) c = 2 . 29. The obtained simulation results are shown in Fig. 10. The number of samples was 100 and the simulation times considered here were the same as for the case c = 10. In the top plot we see that U$_{q,L}$ for different system sizes intersects very near the theoretically predicted critical temperature. The inset of the top plot shows the average overlap order parameter that decreases with system size above the spin-glass critical temperature. In the lower FIG. 7: (Color online) Simulation results for the magneti- zation probability distribution, P ( m ), for the model with c = 10, J$_{0}$ = − 0 . 9 and J = 0 . 5 at d$_{0}$ = 1. In (a) we plot curves for L = 8192 (solid line), L = 4096 (dotted line) at T = 2 . 05 and L = 2048 (dashed line) at T = 2 and in (b) we plot the corresponding data for L = 8192 and L = 4096 at T = 1 . 86 and L = 2048 at T = 1 . 83.