10 2.1 2.2 2.3 2.4 2.5 0 0.1 0.2 0.3 T U q 1 2 3 0 0.5 T < | q | > 0 1 2 3 4 0 5 10 T ˜ χ 0 2 4 0 0.05 0.1 T < | m | > 1.5 2 2.5 3 0 0.5 1 (a) T < | q | > 1.5 2 2.5 3 10 0 10 2 (b) T χ SG 1.5 2 2.5 3 −4 −2 0 (c) T U q FIG. 10: (Color online) Results for the model with c = 4 . 5, J$_{0}$ = − 0 . 9 and J = 1 at d$_{0}$ = 1. In the top plot we show the temperature dependence of the Binder cumulant for the overlap order-parameter. The lines are fits made for each sys- tem size. The vertical line is the location of the predicted spin-glass critical temperature. The inset shows the aver- age values of the overlap order parameter together with the effective field theory prediction for this quantity. In the bot- tom plot we show the temperature dependence of the mag- netic susceptibility. The inset shows the average magneti- zation as a function of temperature. For all the plots the data points correspond to simulations for systems of sides, L = 512( ○ ) , 1024( × ) , 2048( . ) , 4096(+). FIG. 9: (Color online) Simulation results for the average overlap order parameter, < | q | > (a), ˜ χ SG = N< q 2 > (b) and Binder cumulant, U q (c), as a function of tempera- ture for the model with c = 10, J$_{0}$ = − 0 . 9 and J = 0 . 5 at d$_{0}$ = 1. The data points are simulations for L = 512( ○ ) , 1024( × ) , 2048( . ) , 4096(+) , 8192( ⋄ ). ing the numerical two-dimensional Ising model magnetic susceptibility, we can obtain the expected location of the spin-glass phase transition. The result of this calculation is shown in Fig. 11. We stress that theory predicts that the inclusion of an arbitrary small number of long-range shortcuts in the Edwards-Anderson model leads always to a finite temperature phase transition. In fact, in the limit of an infinitesimal addition of short-cuts, the the- ory predicts a P-SG transition at the finite value given by β$_{c}$ a = tanh − $^{1}$[ √ tanh(0 . 44068 ... )] = 0 . 7642 ... ,where 0 . 44068 ... is the critical inverse temperature of the regu- lar two-dimensional Ising model with a unitary positive coupling. This implies that the EA model with short- cuts, in the limit c → 0 $^{+}$, is not equivalent to the orig- inal EA model without short-cuts. In other words, the EA model is not thermodynamically stable under graph noise. plot we see that the magnetic susceptibility is almost independent of system size and it agrees with the theo- retical prediction for temperatures above the spin-glass phase transition while it deviates at lower temperatures. The bimodal Edwards-Anderson model[23] is a disor- dered spin model where the nearest neighbor coupling J$_{0}$ of Ising type spins has the bimodal distribution in (12). Here, we consider the two dimensional square lattice ver- sion of the model with additional long-range couplings. The pure model (without long range shortcuts) is known to show a P-SG phase transition only at zero tempera- ture being the lower critical dimension of the model equal to two [32, 33]. To apply the effective field theory, for each phase Σ =F or SG, we have to consider the pure Ising model magneti- zation (13) and susceptibility (24) calculated by using the definitions of the effective long- and short-range couplings J (Σ) and J (Σ) 0 , from Eqs. (14)-(17), respectively. By us- We made simulations for the EA model with a = 1, J = 1 and c = 1 to compare with the predictions of the theory. The average over disorder was done by considering 100 samples. We neglected the first 10 5 MCS/N for equilibration and made measurements for 10 6 MCS/N. For c = 1 the expected critical temperature is F. Bimodal Edwards-Anderson model