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<img src='content_image/440601.jpg'>

aspects of the ground-state structure of the 2d random- bond BC model. From Fig. 11, reproduced here from Ref. [58], one can see that approximately at this point (Δ = 1 . 5 , r = 0 . 5 / 1 . 5), the system departs from the ferro- magnetic ground state and an unsaturated ground state is produced, which is further enhanced with vacant sites ( s$_{i}$ = 0) as we increase the disorder strength. In the pres- ence of bond randomness the competition between the ferromagnetic interactions with the crystal field results in a destabilization of the ferromagnetic ground state. De- pending on the realization, weak clusters exist in T = 0 and their points are frozen in the s$_{i}$ = 0 state. This is an interesting subject, which is presently under further consideration in both 2d and 3d by the present authors. The novel behavior illustrated in Fig. 10 appears now as a consequence of the onset of the unsaturated ground state at (Δ = 1 . 5 , r = 0 . 5 / 1 . 5) which is thus related with the critical behavior of the 2d random-bond BC model. The presented calculation of the ground states has be carried out in polynomially bounded computing time by mapping the system into a network and searching for a minimum cut by using a maximum flow algorithm (see for instance Ref. [60]) and can be easily extended to large lattices and also to the 3d BC random-bond model. 

By carrying out an extensive two-stage Wang-Landau entropic sampling of both the pure and the random- bond 2d Blume-Capel model we have produced phase diagram points for several disorder strengths. Also for the pure model we found an excellent coincidence of our points with those of Beale [48]. For a particular disorder strength ( r = 0 . 75 / 1 . 25) we found that, as a result of the enhancement of ferromagnetic order, the 

## VI. CONCLUSIONS 

<img src='content_image/440600.jpg'>

phase diagram of the random version crosses that of the pure model at approximately the point (Δ$_{cross}$ , T$_{cross}$ ) = (1 . 66(2) , 1 . 02(3)), well before the tricritical point. 

The critical properties of the pure model were com- pared and contrasted to those of the random model and for moderate values of the crystal field and disorder we found that the Ising universality scenario (with possible logarithmic corrections) is well obtained in the case of the random version. Furthermore, accepting in this range of couplings the assumption of the double logarithmic scenario for the specific heat we estimated multicritical points, where the 2d random-bond (BC) Ising universal- ity class meets the strong-disorder universality class. 

The behavior of the strong disorder regime was also critically discussed. The case Δ = 1 . 5 and r = 0 . 5 / 1 . 5 was extensively studied and based on the observed be- havior and on the ground-state observations we suggested that this case, most likely, lies on the frontier between the weak- and strong-disorder universality classes, suffering exceptionally strong finite-size effects and a strong lack of self-averaging. 

In conclusion, the present paper investigated some difficult and important aspects of the random-bond 2d 

