content_image/388618.jpg content_image/388619.jpg content_image/388620.jpg content_image/388623.jpg content_image/388616.jpg content_image/388617.jpg content_image/388621.jpg content_image/388622.jpg content_image/388624.jpg
and 

In particular, for the susceptibility ˜ χ (Σ) of the random model we have: 

<img src='content_image/388618.jpg'>

where ˜ χ$_{0}$ stands for the susceptibility χ$_{0}$ of the pure model divided by β (we will adopt throughout this di- mensionless definition of the susceptibility) and similarly for the random model. For the case Σ =F without disor- der ( dµ ( J $^{′}$) = δ ( J ′ − J ) dJ ′ and dµ$_{0}$ ( J$_{0}$ ) = δ ( J$_{0}$ − a ) dJ$_{0}$ ), Eq. (24) was already derived in [20] by series expansion techniques at zero field ( h = 0) in the P region (where m = 0). 

Among all the possible stable solutions of Eqs. (13), in the thermodynamic limit, for both Σ=F and Σ=SG, the true solution ¯ m $^{(Σ)}$, or leading solution, is the one that minimizes L (Σ) where 

<img src='content_image/388619.jpg'>

f$_{0}$ ( βJ$_{0}$, βh ) being the free energy density in the thermo- dynamic limit of the pure model with coupling J$_{0}$ and in the presence of an external field h , at inverse tempera- ture β . A necessary condition for a solution m (Σ) to be the leading solution is the stability condition: 

<img src='content_image/388620.jpg'>

For the localization and the reciprocal stability be- tween the F and SG phases we remind the reader to Sec. IIID of [12]. We recall however that, at least for lattices L$_{0}$ having only loops of even length, the stable P region is always that corresponding to a P-F phase diagram, so that in the P region the correlation functions must be calculated only through Eqs. (19) and (22). 

The inverse critical temperature β (Σ) c is solution of the following exact equation 

<img src='content_image/388623.jpg'>

where β (Σ) c 0 is the inverse critical temperature of the pure model with coupling J (Σ) 0 . When J$_{0}$ > 0, the constrain in Eq. (27) ensures the uniqueness of the solution. However, if J$_{0}$ < 0, Eq. (27) in general admits either 0 or at least 2 solutions (in principle also 4, 6, etc...). 

We end this section by stressing that this method is exact in all the P region and, at least for second-order phase transitions, provides the exact critical surface, be- havior and percolation threshold, and that, in the ab- sence of frustration, the order parameters m (Σ) become 

<img src='content_image/388616.jpg'>

<img src='content_image/388617.jpg'>

exact also in the limit c → 0 $^{+}$, in the case of second- order phase transitions, and in the limit c → ∞ (see Sec. IIIC of [12]). Note also that the order parameters m $^{(Σ)}$, and then the correlation functions, are by construction always exact in the zero temperature limit. 

## IV. SIMULATIONS AND COMPARISON WITH THE THEORY FOR GIVEN COUPLINGS 

The Monte-Carlo simulations presented in this work were made using a local spin-flip dynamics with a Metropolis acceptance probability [24]. 

Throughout this work we estimate the susceptibility, in the P phase by ˜ χ = N 〈 m $^{2}$〉 , and in the ferromagnetic phase by ˜ χ = N ( 〈 m $^{2}$〉 − 〈 | m |〉 $^{2}$) , where m = 1 N ∑ N i $_{=1}$σ$_{i}$ is the magnetization of the system and N = L d$_{0}$ is the total number of spins in the lattice of side L . The Binder cumulant[25], defined by 

was used to locate the critical points. The cumulants U$_{L}$ and U$_{L}$ ′ , for two systems of different sides L and L $^{′}$, plotted as a function of temperature, cross at the critical point at a value, U ∗ that characterizes the universality class of the model. 

To study spin-glass phases we calculate the overlap order-parameter, q = ∑ N i =1 σ (1) i σ (2) i obtained from two replicas of the system with spins σ (1) i and σ (2) i . The ob- served distribution of the values of q is measured for a given realization of the disorder which corresponds to taking a thermal average. Subsequently, by considering different samples, an average over disorder is done: 

where q$_{j}$ is the value of the overlap parameter at time step j, δ$_{i,j}$ is the Kronecker delta, M is the number of simulation Monte Carlo steps (MCS) after thermal equi- libration is reached, and the bar denotes averaging over disorder. The Binder cumulant for the overlap order pa- rameter can be defined by 

We study the small-world model with the distribution of random bonds defined in Eq. (9) and a fixed positive long-range coupling constant as in Eq. (10). We start in subsection IV A to study the ferromagnetic case with J$_{0}$ > 0 and the location of the critical points for differ- ent values of c and d$_{0}$ = 1 , 2 and 3. In subsection IV B 

<img src='content_image/388621.jpg'>

<img src='content_image/388622.jpg'>

<img src='content_image/388624.jpg'>

