3 B$_{min}$ = a$_{B}$$_{i}$ + b$_{B}$$_{i}$/N + ... (14) C = C$_{reg}$ + L − $^{α/ν}$f$_{C}$ ( x )[1 + ... ] , (15) [ < | m | > ]$_{av}$ = L − $^{β/ν}$f$_{m}$ ( x )[1 + ... ] , (16) χ = L − $^{γ/ν}$f$_{χ}$ ( x )[1 + ... ] , (17) d ln[ < | m | p > ]$_{av}$ dK = L 1 $^{/ν}$f$_{p}$ ( x )[1 + ... ] , (18) 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. 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 In order to go further in our analysis we also com- 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. FIG. 1: Fourth-order Binder cumulant as a function of K for several values of the system size N = 500 , 1000 , 2000 , 4000 , 8000 and 16000. FIG. 2: Log-log plot of the maxima of the logarithmic deriva- tive d ln[ < | m | $^{p}$> ] dK versus the lattice size L = N 1 / 2 for p = 1 (circle) and p = 2 (square). The solid lines are the best linear fits. III. RESULTS AND DISCUSSION A. undirected random two-dimensional lattice 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 0.340 0.345 0.350 0.355 0.360 0.365 0.370 K 0.50 0.55 0.60 0.65 0.70 U 4 500 16000 3.0 3.5 4.0 4.5 5.0 ln L 3.5 4.0 4.5 5.0 5.5 6.0 Ln(d(ln< | m | p >)/dk) max p=1 p=2