content_image/434830.jpg content_image/434829.jpg
<img src='content_image/434830.jpg'>

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. 

## F. Bimodal Edwards-Anderson model 

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- 

<img src='content_image/434829.jpg'>

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. 

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 

