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plied in the refinement WL levels j = j$_{i}$, . . . , j$_{i}$ + 3 (typ- ically j$_{i}$ = 21), where we usually test both an one-range (one-R) or a multi-R approach with large energy inter- vals. In the case of the one-R approach we have found very convenient and in most cases more accurate to fol- low the Belardinelli and Pereyra [50] adjustment of the WL modification factor according to the rule ln f ∼ t $^{$_{−}$1}$. Finally, it should be also noted that by applying in our scheme a separate accumulation of histogram data in the starting multi-R stage (in the wide energy subspace) of- fers the opportunity to inspect the behavior of all basic thermodynamic functions in an also wide temperature range and not only in the neighborhood of the finite-size anomalies. The approximation outside the dominant en- ergy subspace is not of the same accuracy with that of the restricted dominant energy subspace but is good enough for the observation of the general behavior and provides also a route of inspecting the degree of approximation. 

The above described numerical approach was used to estimate the properties of a large number of 100 bond disorder realizations, for lattice sizes L = 20 − 100 for all crystal fields and disorder strengths used in this paper, with the exception of the case Δ = 1 . 5 and r = 0 . 5 / 1 . 5, where 500 disorder realizations were simulated for lat- tice sizes L = 20 − 140. For reference and contrast, the pure-system’s properties were also obtained by the same implementation, simulating in each case at least 30 in- dependent runs. We close this outline of our numerical scheme with some comments concerning statistical er- rors and disorder averaging. Even for the larger lattice size studied here ( L = 100), and depending on the ther- modynamic parameter, the statistical errors of the WL method were found to be of reasonable magnitude and in some cases to be of the order of the symbol sizes, or even smaller. This was true for both the pure version and the individual random-bond realizations. However, since the nature of the present study is qualitative, not aiming to an accurate exponent estimation, the WL errors will not be presented in our figure illustrations. We also note that, for the random-bond version mainly the averages over the disorder realizations, denoted as [ . . . ]$_{av}$, will be considered in the text and their finite-size anomalies, de- noted as [ . . . ] ∗ $_{av}$, will be used in our FSS attempts. Due to very large sample-to-sample fluctuations, mean values of individual maxima ([ . . . ∗ ]$_{av}$) have not been used in this study except for illustrative purposes, as in the case Δ = 1 . 5, r = 0 . 5 / 1 . 5 presented in our last Section. 

## III. PHASE DIAGRAMS: PURE AND RANDOM-BOND 2D BLUME-CAPEL MODELS 

This Section presents, as mentioned in the introduc- tion, phase diagram points for the pure and random-bond models and in the case of the pure model compare the corresponding phase diagram points with the existing es- timates in the literature. This gives also the opportunity to observe the reliability of our numerical approach. Fol- 

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lowing the practice of our earlier paper [37], we estimate phase diagram points by fitting our data to the expected power-law shift behavior T = T$_{c}$ + bL $_{−}$1 /ν of several pseu- docritical temperatures. The traditionally used specific heat and magnetic susceptibility peaks, as well as, the peaks corresponding to the following logarithmic deriva- tives of the powers n = 1 , 2 , 4 of the order parameter with respect to the inverse temperature K = 1 /T [51], 

and the peak corresponding to the absolute order- parameter derivative 

will be implemented for a simultaneous fitting attempt. 

Such simultaneous fitting attempts are presented in Fig. 1. In particular Fig. 1(a) presents the shift be- havior for the random-bond 2d BC model in the weak 

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