4 B. directed random two-dimensional lattice FIG. 3: Plot of the logarithm of the modulus of the magne- tization at the inflection point as a function of the logarithm of L = N 1 / $^{2}$. The solid line is the best linear fit. FIG. 4: Log-log plot of the susceptibility maxima χ$_{max}$ as a function of the logarithm of L = N 1 / $^{2}$. The solid line is the best linear fit. FIG. 6: Energetic Binder cumulant as a function of the re- duced temperature K for q = 0 . 1 and lattice sizes N = 250 , 500 , 1000 , 2000 , 4000 , 8000, and 16000 from bottom to top. FIG. 5: Magnetization as a function of the temperature, for N = 16000 sites. It is clear a second-order phase transition for q = 0 . 1, and a first-order phase transition for q = 0 . 9, respectively. Binder cumulant as a function of the reduced tempera- ture K for q = 0 . 1, and different lattice sizes, is displayed in the Figure 6, where a characteristic second-order phase transition is observed. On the other hand, in the Figure puted the modulus of the magnetization at the inflection point and the maximum of the magnetic susceptibility. The logarithm of these quantities as a function of the logarithm of L are presented in Figures 3 and 4, respec- tively. A linear fit of these data gives β/ν = 0 . 135(9) from the magnetization and γ/ν = 1 . 751(4) from the susceptibility which should be compared to β/ν = 0 . 125 and γ/ν = 1 . 75 obtained for a regular 2 D lattice. One can see that for this undirected random lattice we get the same universal critical behavior as the regular lat- tice model, in the same way as has been reported for the diluted spin-1 / 2 Ising model $^{32}$. In Fig. 5 we show the dependence of the magnetization as a function of the temperature, obtained from simula- tions on directed random two-dimensional lattice, with N = 16000 sites, and two values of probability q , namely q = 0 . 1 for second-order phase transition and q = 0 . 9 for first-order phase transition, respectively. The energetic 7 we display the same plot as in Figure 6, but now for q = 0 . 4, where we can see that a first-order transition is present for this value of the rewiring probability. In Fig.8, the difference 2 / 3 − B$_{min}$ is shown as a func- tion of the parameter 1 /N for different probabilities q . 3.0 3.5 4.0 4.5 5.0 Ln L −0.9 −0.8 −0.7 −0.6 −0.5 Ln(< | m | >) inf 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 Temperature 0.0 0.2 0.4 0.6 0.8 1.0 Magnetization q=0.1 q=0.9 0.28 0.30 0.32 0.34 K 0.52 0.56 0.60 0.64 0.68 B(K) q=0.1 250 16000 3.0 3.5 4.0 4.5 5.0 Ln L 1.2 2.2 3.2 4.2 5.2 Ln  $_{χ}$max