7 GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s49> GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s83> GLYPH<s112> GLYPH<s101> GLYPH<s99> GLYPH<s105> GLYPH<s102> GLYPH<s105> GLYPH<s99> GLYPH<s32> GLYPH<s72> GLYPH<s101> GLYPH<s97> GLYPH<s116> GLYPH<s76> GLYPH<s32> GLYPH<s80> GLYPH<s117>GLYPH<s114> GLYPH<s101> GLYPH<s32> GLYPH<s114> GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s48>GLYPH<s46> GLYPH<s56>GLYPH<s53>GLYPH<s32> GLYPH<s47>GLYPH<s32> GLYPH<s49>GLYPH<s46> GLYPH<s49>GLYPH<s53> GLYPH<s32> GLYPH<s114> GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s48>GLYPH<s46> GLYPH<s55>GLYPH<s53>GLYPH<s32> GLYPH<s47>GLYPH<s32> GLYPH<s49>GLYPH<s46> GLYPH<s50>GLYPH<s53> GLYPH<s49> GLYPH<s46>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<s50> GLYPH<s49> GLYPH<s46>GLYPH<s54> GLYPH<s50> GLYPH<s46>GLYPH<s48> GLYPH<s50> GLYPH<s46>GLYPH<s52> GLYPH<s50> GLYPH<s46>GLYPH<s56> GLYPH<s51> GLYPH<s46>GLYPH<s50> GLYPH<s51> GLYPH<s46>GLYPH<s54> GLYPH<s50> GLYPH<s46>GLYPH<s56> GLYPH<s51> GLYPH<s46>GLYPH<s50> GLYPH<s51> GLYPH<s46>GLYPH<s54> GLYPH<s52> GLYPH<s46>GLYPH<s48> GLYPH<s52> GLYPH<s46>GLYPH<s52> GLYPH<s52> GLYPH<s46>GLYPH<s56> GLYPH<s50> GLYPH<s51> GLYPH<s52> GLYPH<s53> GLYPH<s54> GLYPH<s82>GLYPH<s97>GLYPH<s110> GLYPH<s100> GLYPH<s111> GLYPH<s109> GLYPH<s32>GLYPH<s77>GLYPH<s111> GLYPH<s100> GLYPH<s101>GLYPH<s108>GLYPH<s58>GLYPH<s32>GLYPH<s114>GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s48> GLYPH<s46>GLYPH<s55> GLYPH<s53> GLYPH<s32>GLYPH<s47>GLYPH<s32>GLYPH<s49> GLYPH<s46>GLYPH<s50> GLYPH<s53> GLYPH<s32> GLYPH<s32> GLYPH<s91> GLYPH<s67> GLYPH<s93> GLYPH<s42> GLYPH<s97> GLYPH<s118> GLYPH<s108>GLYPH<s110>GLYPH<s32>GLYPH<s40>GLYPH<s108>GLYPH<s110>GLYPH<s32>GLYPH<s76> GLYPH<s41> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s91>GLYPH<s67>GLYPH<s93> GLYPH<s42> GLYPH<s97>GLYPH<s118> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s67> GLYPH<s49> GLYPH<s43>GLYPH<s67> GLYPH<s50> GLYPH<s32>GLYPH<s108>GLYPH<s110> GLYPH<s40>GLYPH<s108>GLYPH<s110> GLYPH<s32>GLYPH<s76>GLYPH<s41> GLYPH<s40>GLYPH<s98> GLYPH<s41> GLYPH<s80>GLYPH<s117> GLYPH<s114>GLYPH<s101>GLYPH<s32>GLYPH<s77>GLYPH<s111> GLYPH<s100> GLYPH<s101>GLYPH<s108> GLYPH<s67> GLYPH<s42> GLYPH<s61>GLYPH<s32>GLYPH<s67> GLYPH<s49> GLYPH<s43>GLYPH<s67> GLYPH<s50> GLYPH<s32>GLYPH<s108>GLYPH<s110> GLYPH<s32>GLYPH<s76> GLYPH<s40>GLYPH<s97>GLYPH<s41> GLYPH<s32> GLYPH<s67> GLYPH<s42> GLYPH<s108>GLYPH<s110>GLYPH<s32>GLYPH<s76> GLYPH<s32> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s49> GLYPH<s32> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s49> GLYPH<s46>GLYPH<s53> GLYPH<s32> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s49> GLYPH<s46>GLYPH<s55> GLYPH<s53> GLYPH<s32> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s49> GLYPH<s46>GLYPH<s57> GLYPH<s32> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s49> GLYPH<s46>GLYPH<s57> GLYPH<s53> FIG. 3: Behavior of the random-bond 2d BC model at Δ = 1 . 975 from Ref. [37]. Illustration of the divergence of the specific heat of the pure model (first-order regime) and the clear saturation of the specific heat for the random-bond (open symbols) 2d BC model for two disorder strengths in a log-log scale. refutably distinguish between a double-logarithmic diver- gence and a very weak power-law divergence, the theoret- ically well-grounded double-logarithmic scenario applies very well [37] and this fact can be observed also now for more values of Δ in Fig. 4(b). There are some further features one can observe from Fig. 4. First, for the pure model, and in the range Δ > 1 . 9, we observe a sud- den change in the behavior of the specific heat peaks as we approach the tricritical point which is apparently a strong cross-over effect. For the random version and the same values of Δ no such strong effects are noticeable and most probably the general softening effects of bond randomness extends also to the expected cross-over phe- nomena between the two different universality classes of the random 2d BC model. From Fig. 4(b) the slopes of the double-logarithmic fittings appear to obey a rather sensible decreasing tendency from which we try in the next Section to locate the frontier of strong-disorder uni- versality class. of Δ = 0 − 1 . 95 are used. However, even so, the com- parison of the two panels of Fig. 5, points out the much better limiting behavior of the random version towards the expected Ising value γ/ν = 1 . 75. Therefore, we may convincingly suggest that, for moderate values of crystal field and disorder, the Ising universality scenario (with possible logarithmic corrections) is well obtained in the disordered case. A second interesting comparison follows now in Fig. 5, where again Fig. 5(a) presents the FSS of the suscepti- bility maxima for the pure model, whereas Fig. 5(b) cor- responds to the random-bond version at r = 0 . 75 / 1 . 25. The influence of the exceptionally large fluctuations in the order parameter of the pure model, as we approach the tricritical point, is now reflected in the effective ex- ponent ratio γ/ν , estimated by the simple power law χ ∗ ∼ L $^{γ/ν}$. These large fluctuations are a known pe- culiarity of the pure model near tricriticality [48]. The effective value of the exponent is now closer to the ex- pected value at the tricritical point γ/ν = 1 . 5 [48] than to the value of the Ising universality class γ/ν = 1 . 75. For this reason a separate power-law fitting has been ap- plied for the value Δ = 1 . 95 in Fig. 5(a). Note here that in Fig. 5(a) the simultaneous fitting is applied to the first six values of Δ = 0 − 1 . 9, whereas in Fig. 5(b) all values Figures 6 and 7 illustrate further alternative routes, used commonly in traditional FSS analysis, that pro- vide clear evidence to the above suggestion namely that the weak-disorder version belong to the 2d Ising class for suitable moderate values of disorder and crystal-field coupling. Noteworthy is the fact that the estimation of the critical exponents via the traditional FSS, such as that shown in Figs. 5 - 7 for the random version, yields estimates very close to the expected values, i.e. the ex- ponents of the 2d Ising model. On the other hand, as pointed also earlier, for the pure model, as we approach the tricritical point (Δ = 1 . 95) the effective exponents remind more those expected at the tricritical point than those of the 2d Ising model. This is particularly true for the correlation length’s exponent estimated in Fig. 1(b) by the simultaneous fitting of the six pseudocritical tem- FIG. 4: FSS of the specific heat maxima for the pure and random-bond 2d BC model at the same values of Δ. (a) Pure model: illustration of linear fittings assuming the expected Ising logarithmic divergence. (b) Random model: illustration of linear fittings assuming a double logarithmic divergence. Note the steady fall of the double logarithmic amplitude C 2 .