2 namic parameters used in the traditional FSS analysis of MC data. This study enables us to observe some pecu- liarities of the pure model, due to the onset of tricritical- ity, and compare them with the corresponding behavior of the random model. Furthermore, we try to focus, un- derstand and shed light to the extent of universality of the random-bond 2d BC model with the corresponding random-bond Ising model, for which the scenario of log- arithmic corrections seems to be the strongest option in the current literature [24]. The prediction of the range of such universality is far from trivial and the two regimes (weak and strong) have many dissimilarities which are also reflected in the ground-state structure, as further discussed below. The attempt to estimate the range of the above mentioned universality is accomplished by the novel idea which assumes the truth of the double logarith- mic scenario for the specific heat in a suitable restricted range. This is presented in Sec. V together with some fur- ther crucial observations concerning the behavior of the strong disorder regime, i.e. the regime where the Ising class universality does not apply and the system has a rather complex and yet not understood behavior. Our conclusions are summarized in Sec. VI. large q -cases ( q > 4) for which one could observe and try to classify the above mentioned softening of first-order transitions in 2d models. Another similarly interesting candidate, not yet as much studied in the random-bond version, is the 2d Blume-Capel (BC) model [31, 32]. We may note here that most of the existing literature on the BC model with randomness concerns randomness applied to the crystal field and/or spin glass exchange interac- tions [33–36]. As it is well known, the pure version of the BC model undergoes an Ising-like continuous phase tran- sition to an ordered ferromagnetic phase as the tempera- ture is lowered for crystal-field couplings less than a tri- critical value and a first-order transition for larger values of the crystal-field coupling. Therefore, this model pro- vides also the opportunity to study two different and very interesting topics of the above described effects of disor- der in critical phenomena, namely the double-logarithmic scenario for the specific heat in the regime where the 2d BC model is in the same universality class with the 2d random-bond Ising model and also the softening of the transition in the first-order regime. Recently the present authors [37] have considered this model and provided strong numerical evidence clarifying two of the above mentioned effects induced in 2d systems by bond random- ness. By implementing a two-stage Wang-Landau (WL) approach [37–43], we presented essentially exact informa- tion on the 2d BC model under quenched bond random- ness. In this investigation, we found dramatically dif- ferent critical behaviors of the second-order phase transi- tions emerging from the first- and second-order regimes of the pure BC model and since, these second-order transi- tions were found to have different critical exponents, our study indicated an interesting strong violation of univer- sality [37]. Namely, different sets of critical exponents on two segments of the same critical line appeared to describe the two regimes: still-second-order and ex-first- order. In this paper, we extend our earlier work [37], by im- plementing essentially the same two-stage WL approach (Sec. II) and try to give a more complete picture by con- centrating in the weak (still-second-order) regime and simulate the model for several disorder strengths and many values of the crystal-field coupling. The above statement means that, effectively we will restrict our study to moderate values of the crystal field and moder- ate values of the disorder, intending to observe the fron- tier between the weak- and the strong-disorder universal- ity classes from the disappearance of the expected 2d ran- dom Ising universality class behavior. Thus, in Sec. III we will produce phase diagram points for the random- bond model but also for the pure model, reporting for the pure case a comparison with existing estimates in the lit- erature. More generally, in carrying out this project we have also considered the pure 2d BC model for several values of the crystal-field, in the second-order regime, observing its finite-size scaling (FSS) behavior. Sec. IV presents such a comparative study between random and pure models concerning the behavior of all thermody- where the spin variables s$_{i}$ take on the values − 1 , 0, or +1, < ij > indicates summation over all nearest- neighbor pairs of sites, and J > 0 is the ferromag- netic exchange interaction. The parameter Δ is known as the crystal-field coupling and to fix the tempera- ture scale we set J = 1 and k$_{B}$ = 1. As it is well known, this model has been analyzed, besides the orig- inal mean-field theory [31, 32], by a variety of approx- imations and numerical approaches. These include the real space renormalization group, MC simulations, and MC renormalization-group calculations [44], ϵ -expansion renormalization groups [45], high- and low-temperature series calculations [46], a phenomenological FSS analy- sis using a strip geometry [47, 48], and, finally, a recent two-parameter WL sampling in rather small lattices of linear sizes L ≤ 16 [49]. As mentioned already in the in- troduction the phase diagram of the model consists of a segment of continuous Ising-like transitions at high tem- peratures and low values of the crystal field which ends at a tricritical point, where it is joined with a second segment of first-order transitions between (Δ$_{t}$ , T$_{t}$ ) and (Δ = 2 , T = 0). The (pure) BC model [31, 32] is defined by the Hamil- tonian H$_{p}$ = − J ∑ <ij> s$_{i}$s$_{j}$ + Δ ∑ i s 2 $_{i}$, (1) A. The pure and random-bond Blume-Capel models II. DEFINITION OF THE MODELS AND THE TWO-STAGE WANG-LANDAU APPROACH