GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s53>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s54>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s55> GLYPH<s48>GLYPH<s46>GLYPH<s55>GLYPH<s53> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48>GLYPH<s48> GLYPH<s77> GLYPH<s99> GLYPH<s76> GLYPH<s32>GLYPH<s47>GLYPH<s32> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s48> GLYPH<s46>GLYPH<s49> GLYPH<s50> GLYPH<s53> GLYPH<s50> GLYPH<s40>GLYPH<s52> GLYPH<s41> GLYPH<s40>GLYPH<s97>GLYPH<s41> GLYPH<s32>GLYPH<s47>GLYPH<s32> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s49> GLYPH<s46>GLYPH<s55> GLYPH<s53> GLYPH<s52> GLYPH<s40>GLYPH<s53> GLYPH<s41> GLYPH<s40>GLYPH<s98>GLYPH<s41> GLYPH<s32> GLYPH<s99> GLYPH<s76> FIG. 4: Magnetic exponent ratios of the pure TrIM: FSS in a log-log scale of (a) the order parameter and (b) the magnetic susceptibility at the critical temperature. In both cases linear fittings are applied. R = 1) the following values B = B ( T$_{c}$ ) = 0 . 14185 · · · and B$_{2}$ = B$_{2}$ ( T$_{c}$ ) = − 0 . 15003 · · · . It is of interest at this point to examine the compat- ibility of our numerical data with the above expansion for the case of the critical specific heat data, i.e. the data of the specific heat at the exact critical tempera- ture of the TrIM. In Fig. 3 we consider only the first two terms in the above expansion (i.e. we set also B$_{2}$ = 0) and pay attention in estimating the critical amplitude A$_{0}$ and the constant B -contribution. From the results of the fitting in the total lattice range L = 20 − 200, as also shown in the figure, we observe that the estimated value for the critical amplitude A$_{0}$ = 0 . 4990(9) is very close to the exact value A$_{0}$ = 0 . 499069 · · · [63]. For the first B -coefficient we find the value B = 0 . 138(5) which is in good agreement with the exact result 0 . 14185 · · · . Let us note here that if we try to fit the data of Fig. 3 including also the coefficient B$_{2}$ of the expansion (7) we get the estimates 0 . 4952(59), 0 . 153(16), and − 0 . 20(12) for the critical amplitude A$_{0}$ and the coefficients B and B$_{2}$ , respectively. However, if we fix the values of A$_{0}$ and B to their exact ones, the estimate for B$_{2}$ we get from the fitting is − 0 . 158(11), in excellent agreement with the exact value, whereas if we fix the value of B and B$_{2}$ to their exact ones we get the estimate 0 . 4990(3) for the Figure 5(b) illustrates again the shift behavior of the 6 pseudocritical temperatures of panel (a), estimated now via the second way of averaging discussed in Sec. I, i.e. by taking the average over the individual pseudocriti- We start our analysis of the RBTrIM by presenting the general shift behavior of various pseudocritical tem- peratures of the model. Figure 5(a) illustrates the shift behavior of 7 pseudocritical temperatures defined as the temperature where the corresponding average thermo- dynamic property attains its maximum. The first 6 are as those defined in Sec. II for the corresponding pure model. The last pseudocritical temperature is a newly introduced temperature, defined as the tempera- ture where the ratio R$_{[}$$_{χ}$ ∗ ] av defined in Sec. II, becomes maximum. The solid lines show an excellent simultane- ous power-law fitting attempt of the form (6) giving the value T$_{c}$ = 3 . 4642(52) for the critical temperature of the random model and a value ν = 0 . 997(6) for the critical exponent ν of the correlation length. The fitting shown in Fig. 5(a) has been performed for all lattice sizes and it was very stable when shifting the L -range to larger values. Note also that a simple fitting attempt using only the newly defined pseudocritical temperature de- fined with the help of the sample-to-sample fluctuations gives a value T$_{c}$ = 3 . 4677(47) for the critical tempera- ture and a value ν = 0 . 994(8) for the correlation length’s exponent. These overall estimates for the exponent ν consist a strong indication that the RBTrIM shares the same value of ν as the pure version, thus reinforcing the scenario of logarithmic corrections (strong universality). We now present our numerical results for the random- bond version of the triangular Ising model for disorder strength r = 1 / 3. From simple universality-type theoret- ical arguments, this system is also expected to undergo a second-order transition between the ferromagnetic and paramagnetic phases and in particular it should be also expected that this transition will be in the same univer- sality class as the RBSqIM. IV. RANDOM MODEL critical amplitude A$_{0}$ . Finally, Figs. 4(a) and (b) present our estimations for the magnetic exponent ratios β/ν and γ/ν . For the esti- mation of β/ν we use the values of the order parameter at the exact critical temperature. As shown in panel (a), in a log-log scale, the linear fitting provides the es- timate β/ν = 0 . 1252(4). In panel (b) we show the FSS of the critical susceptibility, also in a log-log scale. The straight line is a linear fitting for L ≥ 20 giving the esti- mate γ/ν = 1 . 754(5). Thus, our results presented in this Section for the pure 2d TrIM model are in excellent agree- ment with the exact results and also with the expected logarithmic expansion of the specific heat [62, 79]. This consists a very strong accuracy test of the proposed two- stage WL entropic sampling in restricted energy spaces. 5