content_image/388522.jpg content_image/388523.jpg content_image/388524.jpg content_image/388525.jpg content_image/388526.jpg content_image/388539.jpg content_image/388540.jpg
On the other hand, if the hypothesis of a second-order phase transition is correct, we should then expect, for large systems sizes, an asymptotic FSS behavior of the form 

where C$_{reg}$ is a regular background term, ν , α , β , and γ are the usual critical exponents, and f$_{i}$ ( x ) are FSS func- tions with x = ( K − K$_{c}$ ) L 1 /ν being the scaling variable, and the brackets [1 + ... ] indicate corrections-to-scaling terms. 

In all cases, we estimated the error bars from the fluctuations among the different realizations. Note that these errors contain both, the average thermodynamic error for a given realization and the theoretical variance for infinitely accurate thermodynamic averages which are caused by the variation of the quenched, random geome- try of the lattices. 

## A. undirected random two-dimensional lattice 

By applying standard re-weighting techniques to each of the R time-series data we first determined the temper- ature dependence of C$_{i}$ ( K ), χ$_{i}$ ( K ),..., i = 1,..., R , in the neighborhood of the simulation point K$_{0}$ . Once the tem- perature dependence is known for each realization, we can easily compute the disorder average, e.g., C ( K ) = ∑ R i $_{=1}$C$_{i}$ ( K ) /R , and then determine the maxima of the averaged quantities, e.g., C$_{max}$ ( K$_{max}$ ) = max$_{K}$C ( K ). The variable R represents the number of replicas in our simulations. 

In order to estimate the critical temperature we calcu- late the second and fourth-order Binder cumulants given by eqs. (7) and (8), respectively. It is well known that these quantities are independent of the system size and should intercept at the critical temperature $^{31}$. In Fig. 1 the fourth-order Binder cumulant is shown as a function of the K for several values of N . Taking the largest lat- tices we have K$_{c}$ = 0 . 3560(5). To estimate U ∗ 4 we note that it varies little at K$_{c}$ so we have U ∗ 4 = 0 . 6058(6). From the second-order cumulant (not shown) we sim- ilarly get K$_{c}$ = 0 . 3561(4) and U ∗ 2 = 0 . 6416(3). One can see that the agreement of the critical temperature 

## III. RESULTS AND DISCUSSION 

<img src='content_image/388522.jpg'>

<img src='content_image/388523.jpg'>

<img src='content_image/388524.jpg'>

<img src='content_image/388525.jpg'>

<img src='content_image/388526.jpg'>

<img src='content_image/388539.jpg'>

is quite good and U ∗ 4 is definitely close to the universal value U ∗ 4 ∼ 0 . 61 for the same model on the regular 2 D lattice. 

The correlation length exponent can be estimated from the derivatives given by eq. (18). Figure 2 shows the maxima of the logarithm derivatives as a function of the logarithm of the lattice size L for p = 1 and p = 2. From the linear fitting one gets ν = 0 . 984(6) ( p = 1) and ν = 0 . 983(5) ( p = 2), which is again next to the regular lattice exponent ν = 1. 

<img src='content_image/388540.jpg'>

In order to go further in our analysis we also com- 

