6 0 0.5 1 0 2 4 w(T) 0 0.5 1 1.5 0 1 0 1 2 T 0 1 2 w(T) 0 1 2 T 0 1 2 b = 0.4 b = 1 b = 2 b = 4 1 2 3 4 5 M 0 1 2 <T> b = 0.02 b = 0.04 b = 0.2 b = 0.4 b = 1 b = 2 b = 4 b = 10 3DAM 0.01 0.1 1 10 b 0 0.5 1 <T>/<T> COE M = 1 M = 2 M = 3 M = 4 M = 5 1 2 3 4 5 M 1 2 3 (a) (b) FIG. 6: (Color online) Black [red (gray)] curves: Conductance probability distribution w ( T ) for the periodic [non-periodic] PBRM model at criticality for some large values of b in the case M = 2. Blue (gray) dashed lines are w ( T ) COE ; the RMT prediction for w ( T ) given by Eq. (9). FIG. 7: (Color online) (a) Average conductance 〈 T 〉 as a func- tion of M for the non-periodic PBRM model at criticality for several values of b . 〈 T 〉 for the 3D Anderson model at critical- ity (3DAM) is also shown. (b) 〈 T 〉 / 〈 T 〉$_{COE}$ as a function of bδ for the periodic PBRM model at criticality for M ∈ [1 , 5]. Insert: δ versus M . δ is obtained from the fitting of Eq. (24) to the 〈 T 〉 vs. b data. Thick full lines correspond to 〈 T 〉 = 0. Blue (gray) dashed lines are (a) the RMT prediction for 〈 T 〉 given by Eq. (10); and (b) one. (decreases) as a function of L for α < 1 ( α > 1 and α = 1). Now, in Figs. 5 and 6 we explore w ( T ) in the case M = 2. As well as in the case M = 1, studied above, here: (i) for small b , ρ (ln T ) is scale invariant with T typ as scaling factor, see Fig. 5; (ii) for b ≥ 4, w ( T ) is well described by Eq. (9), the corresponding RMT prediction, see Fig. 6. However, although ρ (ln T ) for small b and w ( T ) for large b are practically the same for the periodic and non-periodic versions of the PBRM model, they show differences for intermediate values, 0 . 1 < b < 4, as can be seen in Fig. 5(a) and the upper panels of Fig. 6. We point out that for small T , T ≪ T typ , ρ (ln T ) is proportional to ( T/T typ ) $^{2}$, as shown in Fig. 5(c). This behavior is universal for the PBRM model with M = 2; i.e. it is b -independent and valid for the periodic and non- periodic versions of the model. Again, as for the M = 1 case, here for M = 2, the dependence ρ (ln T ) ∝ T 2 in the limit b → ∞ is consistent with the RMT prediction w ( T ) COE ∝ T for T < 1; see Eq. (9). From the results above and since 38 in the region 0 < T < 1, we conclude that for the PBRM model for T ≪ T typ . We argue that the behavior dictated by Eq. (23) can be interpreted in the following way: Since small values of conductance T ≪ 1 mean very strong re- flection, the scattering process is almost direct, i.e. an incident electron is scattered out the system mainly by the interaction with the sites at which the leads are at- tached. Then the electron does not explore the complete sample, not even part of it, and as a consequence it does not realize that the sample is at criticality. That is, for T ≪ 1, the incident electron does not distinguish between a critical system and a random system represented by a full random matrix if the number of attached leads 2 M is not much larger than b ; i.e., when the leads are attached to sites interconnected by Hamiltonian matrix elements located within the bandwidth of the matrix where all el- ements have almost the same variance: 〈 | H$_{ij}$ | $^{2}$〉 ≈ 1 / 2. So, we can use Eq. (22) and as a consequence Eq. (23) too. In fact, we have verified that Eq. (23) is valid up to M = 2 for b ≤ 1, while for b = 10 it describes perfectly the left tail of ρ (ln T ) for all the values of M used in this work (up to M = 5). Our result, given by Eq. (23), is the generalization of the result shown in Ref. [28], where the behavior ρ (ln T ) ∝ T 1 / 2 was reported for the PBRM model in the localized ( α > 1) and delocalized regimes ( α < 1) for M = 1. w ( T ) COE ∝ T M $^{2}$/ 2 − 1 , (22) ρ (ln T ) ∝ T M $^{2}$/ 2 , (23)