content_image/439004.jpg content_image/439013.jpg content_image/439003.jpg content_image/439012.jpg content_image/439011.jpg
<img src='content_image/439004.jpg'>

expected value − 0 . 5; whereas for the susceptibility we ob- tain the exponents 1 . 03(5) , 1 . 06(7) , 0 . 97(2) and 1 . 01(5), also close to the expected result for mean-field behavior. 

## B. Correlation functions above the critical temperature 

From Eq. (18) we see that the effective field theory, in the P region and zero external field, predicts the spin- spin correlation function of the random model C ( F ) to be, in the thermodynamic limit, equal to the correlation function of the pure model calculated at the same tem- perature. In order to check this result we calculated, from simulation, 

where σ$_{0}$ is an arbitrary spin and σ$_{r}$ is one spin at Euclidean distance r from the spin σ$_{0}$ , measured on the lattice L$_{0}$ . We considered the case d$_{0}$ = 2 with J = J$_{0}$ and c = 1 and 2 and the inverse temperatures βJ$_{0}$ = 0 . 1 , 0 . 17571. These temperatures are above the critical temperatures for the two values of c studied. We studied also the correlation function at the critical in- verse temperatures β$_{c}$J$_{0}$ = 0 . 2461 and β$_{c}$J$_{0}$ = 0 . 1897 for c = 1 and c = 2, respectively (see Table I). These calcu- lations were done for system sides L = 8 , 16 , 32 and 64. In Fig. 3 we can see that, as the system size increases, 

<img src='content_image/439013.jpg'>

<img src='content_image/439003.jpg'>

the curves C $^{F}$( r ), for c > 0, approach the data points for c = 0 calculated for a system of size L = 64. Note that, as we approach the critical temperature, the correlation function at large r reaches a finite value that decreases as the system size increases. This finite constant is just due to the finite size of the system and it is predicted by the theory to vanish as 1 /N , in the thermodynamic limit, but at the same time it is responsible for the divergence, with system size, of the susceptibility at the critical point (see Eq. (33) and subsequent comments of Ref. [12]). 

## C. Susceptibility and Magnetization at non-zero external field above the critical temperature 

In the P phase at zero external magnetic field the effec- tive field theory prediction for the susceptibility is exact, consistently with the exact predictions for the critical temperatures. The question remains whether the sus- ceptibility prediction is a good approximation for non- zero field. To verify this we made simulations for the case J$_{0}$ = 0 with a positive long-range coupling (as in Eq. (10)) (in other words the simplest version of Viana- Bray model). For this particular case the magnetization is predicted to be given by the solution of the following equation 

<img src='content_image/439012.jpg'>

and the susceptibility (divided by β ) is given by, 

<img src='content_image/439011.jpg'>

