GLYPH<s48> GLYPH<s52>GLYPH<s48> GLYPH<s56>GLYPH<s48> GLYPH<s49>GLYPH<s50>GLYPH<s48> GLYPH<s49>GLYPH<s54>GLYPH<s48> GLYPH<s50>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s50> GLYPH<s49>GLYPH<s46>GLYPH<s52> GLYPH<s49>GLYPH<s46>GLYPH<s54> GLYPH<s49>GLYPH<s46>GLYPH<s56> GLYPH<s50>GLYPH<s46>GLYPH<s48> GLYPH<s50>GLYPH<s46>GLYPH<s50> GLYPH<s49>GLYPH<s46>GLYPH<s52> GLYPH<s49>GLYPH<s46>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s54> GLYPH<s49>GLYPH<s46>GLYPH<s55> GLYPH<s49>GLYPH<s46>GLYPH<s54> GLYPH<s49>GLYPH<s46>GLYPH<s56> GLYPH<s50>GLYPH<s46>GLYPH<s48> GLYPH<s50>GLYPH<s46>GLYPH<s50> GLYPH<s50>GLYPH<s46>GLYPH<s52> GLYPH<s76> GLYPH<s109>GLYPH<s105>GLYPH<s110> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s54>GLYPH<s48> GLYPH<s32> GLYPH<s32> GLYPH<s91>GLYPH<s67> GLYPH<s42> GLYPH<s93> GLYPH<s97>GLYPH<s118> GLYPH<s108>GLYPH<s110>GLYPH<s32>GLYPH<s40>GLYPH<s108>GLYPH<s110>GLYPH<s32>GLYPH<s76>GLYPH<s41> GLYPH<s76> GLYPH<s109>GLYPH<s105>GLYPH<s110> GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s54>GLYPH<s48> GLYPH<s91> GLYPH<s67> GLYPH<s93> GLYPH<s42> GLYPH<s97> GLYPH<s118> GLYPH<s32> GLYPH<s59> GLYPH<s32> GLYPH<s91> GLYPH<s67> GLYPH<s42> GLYPH<s93> GLYPH<s97> GLYPH<s118> GLYPH<s76> GLYPH<s32> GLYPH<s91>GLYPH<s67>GLYPH<s93> GLYPH<s42> GLYPH<s97>GLYPH<s118> GLYPH<s32> GLYPH<s91>GLYPH<s67> GLYPH<s42> GLYPH<s93> GLYPH<s97>GLYPH<s118> GLYPH<s32> FIG. 7: FSS of the specific heat maxima obtained via the two distinct ways of disorder averaging. The solid and dotted lines are double-logarithmic fittings of the form (9) for lattice sizes in the range L = 60 − 200. The inset shows the data for the case [ C ∗ ] av as a function of the double logarithm of L . The solid line is an excellent linear fitting. Tomita and Okabe, using the probability-changing clus- ter algorithm [48]. As in the pure case, the second alternative estimation of ν is carried out by analyzing the divergency of the log- arithmic derivatives of the order parameter. In Fig. 6(a) we illustrate in a double-logarithmic scale the size depen- dence of the first- (filled squares), second- (filled circles), and fourth-order (filled triangles) logarithmic derivatives (averaged over the individual maxima). The solid lines show linear fittings for the sizes L ≥ 60. In all cases a value ν = 1 is obtained for the critical exponent ν , providing further evidence to the strong universality sce- nario emerged from Fig. 5. Figure 6(b) illustrates our method to evaluate and discuss the stability of the esti- mation for the exponent ν from the scaling behavior of the logarithmic derivatives of panel (a). It shows val- ues of effective exponents ( ν$_{eff}$ ) determined by impos- ing a lower cutoff ( L$_{min}$ ) and applying simultaneous fit- tings in windows ( L$_{min}$ − L$_{max}$ ), where as for the pure case, L$_{max}$ = 200 and L$_{min}$ = 20 , 40 , 60 , 80, and 100 as a function of 1 /L$_{min}$ . The effective estimates show a finite-size effect for small values of the lower cutoff, whereas and for L ≥ 60 a clear trend towards the value ν = 1 of the Ising universality class is obtained. Let us note here that the same picture emerged from the FSS of the disorder-averaged logarithmic derivatives of the form [ ∂ ln 〈 M $^{n}$〉 /∂K ] ∗ av that corresponds to the first way of averaging, but is omitted here for brevity. We should note here that a similar cross-over behavior in the esti- mates of the critical exponent ν has been observed in the case of the 2d site-diluted SqIM by Ballesteros et al. [39] and has been explained as logarithmic corrections. Thus, summarizing our estimates for the critical exponent ν , we feel that it is clear that it maintains the value ν = 1 of the pure case, indicating again the validity of the strong universality scenario. In Fig. 8 we provide estimates for the magnetic expo- nent ratios β/ν and γ/ν of the RBTrIM. In panel (a) we plot the average magnetization at the estimated critical temperature, as a function of the lattice size L in a log- log scale. The solid line is a linear fitting for L ≥ 20 giving within error bars the value of the pure model, i.e. β/ν = 0 . 1253(5) ≈ 0 . 125. Additional estimate for the ratio β/ν can be obtained from the FSS of the derivative of the absolute order parameter with respect to inverse temperature defined in Eq. (4) which is expected to scale as L (1 − β ) /ν with the system size [75]. Thus, in panel (b) of Fig. 8 we plot the data for ∂ 〈| M |〉 /∂K averaged over disorder as a function of L , also in a double-logarithmic scale. The solid line is a linear fitting for the larger lat- tice sizes L ≥ 60, which combined with the value ν = 1, gives an estimate of 0 . 1247(4) for the ratio β/ν . Finally, In fact, the double-logarithmic fitting is shown in the main panel, whereas in the corresponding inset of Fig. 7 the data of [ C ∗ ]$_{av}$ are plotted as a function of ln (ln L ). The solid line shown is an excellent linear fit for L ≥ 60. Let us now give some details on the qual- ity of the applied fittings. We used the following sets of data points ( L$_{min}$ − L$_{max}$ ), with L$_{max}$ = 200 and L$_{min}$ = 20 , 40 , 60 , 80, and 100. The quality of the fittings indicated a very good trend for the values of χ $^{2}$/DoF for the double logarithmic fittings (9) in the range: 0 . 2 − 0 . 7 and for both sets of data points. However, a strong relia- bility test in favor of the logarithmic corrections scenario is provided by the stability of the coefficient C$_{2}$ , for both [ C ] ∗ av ( C$_{2}$ ≈ 1 . 43(5)) and [ C ∗ ]$_{av}$ ( C$_{2}$ ≈ 1 . 48(4)) data. On the other hand, the estimated values of the exponent α/ν of the power law (10), for both [ C ] ∗ av and [ C ∗ ]$_{av}$, fluctu- ate in the range [ − 0 . 12(9) , − 0 . 05(6)] (as L$_{min}$ increases) with the fitting procedure becoming rather unstable as we move to larger values of L$_{min}$ . The conclusion is that our numerical data are more properly described by the double logarithmic form (9), in agreement with the MC findings of Selke et al. [43] and Ballesteros et al. [39] for the site-diluted SqIM and also with those of Wang et al. [28] for the strong disorder regime ( r = 1 / 4 and r = 1 / 10) of the RBSqIM. Although it is rather difficult to numerically distinguish between the above scenarios, our detailed fitting at- tempts indicated that the double-logarithmic scenario [Eq. (9)] applies better to the numerical data and this is generally true for both [ C ] ∗ av and [ C ∗ ]$_{av}$ data. or a simple power law We continue the presentation of our results by showing in Fig. 7 the FSS of the specific heat maxima averaged over disorder: [ C ] ∗ av (up filled triangles) and [ C ∗ ]$_{av}$ (down open triangles). Using these data for the larger sizes L ≥ 60, we tried to observe the quality of the fittings, assuming a double-logarithmic divergence of the form 7 [ C ] ∗ $_{av}$; [ C ∗ ]$_{av}$ ∼ C$_{1}$ + C$_{2}$ ln (ln L ) , (9) [ C ] ∗ $_{av}$; [ C ∗ ]$_{av}$ ∼ C ∞ + C$_{3}$L $^{α/ν}$. (10)