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the pure model and for the random version the disor- der strength r = 0 . 75 / 1 . 25 was chosen in both cases. The strong violation of universality observed appeared to be the result of the softening of the first-order tran- sition due to bond-randomness. Specifically, it was con- cluded that the new strong-disorder universality class is well described by a correlation length exponent in the range ν = 1 . 30(6) − 1 . 35(5), and exponent ratios γ/ν and β/ν very close to the Ising values 1 . 75 and 0 . 125, re- spectively [37]. The above weak universality [16, 53, 54] seems to be valid between the Ising-like continuous tran- sitions of the random-bond 2d BC model for small values of Δ (Δ = 1 . 0 in Ref. [37]) and continuous transitions belonging to the strong-disorder universality class. 

Therefore, the strong-disorder universality class may be characterized by the above distinct value of the corre- lation length exponent and a strong saturation of the spe- cific heat. Qualitatively this saturating behavior is quite instructive and for illustrative reasons is reproduced here in Fig. 3. This figure contrasts, at Δ = 1 . 975, the specific heat’s finite-size behavior of the pure 2d BC model (first- order regime) and two disordered cases corresponding to disorder strengths r = 0 . 85 / 1 . 15 and r = 0 . 75 / 1 . 25. The saturation of the specific heat is very clear in both cases of the disorder strength. 

It is of interest to point out here that these findings for the strong-disorder universality class appear to be fully compatible with the classification of phase transitions in disordered systems proposed recently by Wu [55]. Ac- cording to this classification the strong-disorder transi- tion is expected to be inhomogeneous and percolative with an expected exponent of the order ν = 1 . 34 [56]. Furthermore, it has been suggested to us by Wu [57], that the strong lack of self-averaging of this transition stems from the above properties. This violation of self- 

averaging, together with the strong finite-size effects, make the systematic MC approach of the strong-disorder regime very demanding, if not impractical. On the other hand, the weak regime (or Ising universality regime) suf- fers a much weaker lack of self-averaging, by at least a factor of ∼ 12 [37], and a smooth behavior is observed at moderate lattice sizes. Thus, aiming here to observe, even approximately, the extent of the involved universal- ity classes we carried out our study at moderate values of the crystal field and disorder, and found a behavior quite convincing from which the frontier of the strong universality class can be estimated by observing the dis- appearance of the expected 2d random Ising universality class. 

## B. Pure and random-bond 2d BC model: Range of universality with the 2d Ising model 

Let us now proceed with the analysis of our numerical data for the disorder strengths and crystal fields given in Table I and observe and contrast their FSS behav- ior with that of the pure model. Starting this com- parative study with the FSS of the specific heat max- ima (using for the random-bond version at the strength r = 0 . 75 / 1 . 25 the corresponding quantity averaged over disorder, i.e. [ C ] ∗ $_{av}$), we present in Fig. 4 fitting at- tempts for the same range of Δ for the pure model [Fig. 4(a)] and the random-bond version [Fig. 4(b)]. As indicated by the scales in the x-axis and the functions in the corresponding panels, the expected Ising logarith- mic divergence has been assumed for the pure model C ∗ = C 1 + C 2 ln L , whereas the double-logarithmic di- vergence [ C ] ∗ av = C 1 + C 2 ln(ln L ) has been assumed for the random version. Although, it is very difficult to ir- 

