9 GLYPH<s48>GLYPH<s46> GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s53> GLYPH<s49>GLYPH<s46> GLYPH<s48> GLYPH<s49>GLYPH<s46> GLYPH<s53> GLYPH<s50>GLYPH<s46> GLYPH<s48> GLYPH<s50>GLYPH<s46> GLYPH<s53> GLYPH<s48>GLYPH<s46> GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s49> GLYPH<s48>GLYPH<s46> GLYPH<s50> GLYPH<s48>GLYPH<s46> GLYPH<s51> GLYPH<s48>GLYPH<s46> GLYPH<s52> GLYPH<s48>GLYPH<s46> GLYPH<s53> GLYPH<s48>GLYPH<s46> GLYPH<s54> GLYPH<s48>GLYPH<s46> GLYPH<s55> GLYPH<s48>GLYPH<s46> GLYPH<s56> GLYPH<s48>GLYPH<s46> GLYPH<s57> GLYPH<s116> GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s49>GLYPH<s46>GLYPH<s57>GLYPH<s54>GLYPH<s40>GLYPH<s49>GLYPH<s41> GLYPH<s67> GLYPH<s50> GLYPH<s80> GLYPH<s117>GLYPH<s114>GLYPH<s101>GLYPH<s32>GLYPH<s77> GLYPH<s111>GLYPH<s100>GLYPH<s101>GLYPH<s108> GLYPH<s67> GLYPH<s50> GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s99>GLYPH<s111>GLYPH<s110>GLYPH<s115>GLYPH<s116>GLYPH<s32>GLYPH<s40> GLYPH<s116> GLYPH<s45> GLYPH<s41> GLYPH<s120> GLYPH<s50>GLYPH<s48> GLYPH<s52>GLYPH<s48> GLYPH<s54>GLYPH<s48> GLYPH<s56>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s53> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s53> GLYPH<s50>GLYPH<s48> GLYPH<s50>GLYPH<s53> GLYPH<s51>GLYPH<s48> GLYPH<s51>GLYPH<s53> GLYPH<s40> GLYPH<s49>GLYPH<s45> GLYPH<s41> GLYPH<s32> GLYPH<s47>GLYPH<s32> GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s48>GLYPH<s46> GLYPH<s56>GLYPH<s55>GLYPH<s50>GLYPH<s40> GLYPH<s54>GLYPH<s41> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s47>GLYPH<s32> GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s48>GLYPH<s46> GLYPH<s49>GLYPH<s50>GLYPH<s56>GLYPH<s40> GLYPH<s54>GLYPH<s41> 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<s91> GLYPH<s32> GLYPH<s60> GLYPH<s124>GLYPH<s77> GLYPH<s124>GLYPH<s62> GLYPH<s32> GLYPH<s47> GLYPH<s32> GLYPH<s32> GLYPH<s75> GLYPH<s93> GLYPH<s42> GLYPH<s97> GLYPH<s118> GLYPH<s76> GLYPH<s32> GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s48>GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s48>GLYPH<s46> GLYPH<s53> GLYPH<s32> GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s49> FIG. 8: An approximate estimation of the tricritical value of the crystal field by fitting the decreasing logarithmic ampli- tudes C 2 of the pure model at suitable values of the crystal- field coupling to a suitable power law shown in the figure. FIG. 7: Simultaneous fitting to a simple power law of the av- eraged peaks corresponding to the absolute order-parameter derivative for three values of Δ and disorder strength r = 0 . 6. The fitting provides an estimate of the exponent (1 − β ) /ν and as indicated in the panel an alternative estimate for the exponent ratio β/ν , by assuming ν = 1. lack of self-averaging one may also expect large finite-size and cross-over effects. However, such cases are elucida- tory not only for observing the intrinsic difficulties but also for giving us the opportunity to reflect on possible links with basic properties of the system as for instance the ground-state structure. frontier of the strong-disorder universality class moves to lower values of the crystal field, namely: ( r = 0 . 9 / 1 . 1, Δ$_{sd}$ = 1 . 963(8)), ( r = 0 . 75 / 1 . 25, Δ$_{sd}$ = 1 . 955(5)), and ( r = 0 . 6 / 1 . 4, Δ$_{sd}$ = 1 . 879(12)). This behavior seems sensible and in our opinion reflects the competition be- tween the ferromagnetic random interactions with the crystal field, giving a kind of destabilization of the usual Ising-like ferromagnetic order. A similar ground-state reflection of this competition is a ground-state structure of unsaturated ferromagnetic ground states discussed in our recent paper [58]. Conversely, as disorder strength is decreased, Δ$_{sd}$ approaches Δ$_{t}$, as expected. Thus, Fig. 10 illustrates two important characteristics of the case Δ = 1 . 5 and r = 0 . 5 / 1 . 5. In particular, Fig. 10(a) shows the huge sample-to-sample fluctuations in all the pseudocritical temperatures used in this pa- per. The simulation here was extended to larger lattices ( L = 20 − 140) and the number of realizations studied was 500, to be compared with 100 realizations studied in previous Sections (weak-regime). Besides the enor- mous fluctuations, the number of 500 realizations looks quite insufficient and strong finite-size effects, reflected as expected in the shift behavior of the system, produce a completely unsettled behavior. Figure 10(b) illustrates an unusual deviating behavior between the finite-size be- havior of the maxima of the averaged specific heat curves ([ C ] ∗ $_{av}$) and the finite-size behavior of the average of in- dividual maxima ([ C ∗ ]$_{av}$), which as shown suffer large sample-to-sample fluctuations. The behavior here resem- bles in many aspects the well known and still challeng- ing specific heat behavior of the 3d random-field Ising model [42]. However, a very clear tendency of [ C ] ∗ av for a saturating behavior is observed in Fig. 10(b) and we may speculate that this saturating behavior is the correct asymptotic behavior for both maxima shown in Fig. 10(b), although the behavior of [ C ∗ ]$_{av}$ will settle down only in very large lattice sizes, when the influence of the strong finite-size effects on the individual maxima will diminish. The above illustrations open the possibility that the case Δ = 1 . 5 and r = 0 . 5 / 1 . 5 is very close to the fron- tier of the strong-universality class. This is in accor- dance with the trend observed in the previous Section and can be further supported by reproducing here some As pointed out earlier, the strong lack of self-averaging, together with possible strong finite-size effects, make the MC approach to the strong-disorder regime a very difficult task. The self-averaging properties along the two segments (ex-first-order and still-second-order) of the critical line were observed and discussed in Ref. [37]. It was shown in this paper that the usual finite-size mea- sure [59] of relative variance R$_{X}$ = V$_{X}$/ [ X ] 2 $_{av}$, where V$_{X}$ = [ X $^{2}$]$_{av}$ − [ X ] 2 av (and X = χ ∗ is the susceptibil- ity maxima), exhibits lack of self-averaging in both cases of marginal second-order transition and the transition in the strong-disorder or ex-first-order regime. In particu- lar, it was shown [37] that the case studied (Δ = 1 . 975) of the ex-first-order segment gives a much larger effect when compared with the still-second-order regime at Δ = 1 and the same disorder strength r = 0 . 75 / 1 . 25 by a fac- tor of ∼ 12. Therefore, phase diagram points very close to the frontier of the strong-disorder regime may be the worst cases to study, because, besides the very strong B. Strong disorder regime: The case Δ = 1 . 5 , r = 0 . 5 / 1 . 5 and general observations