2 variables drawn from a normal distribution with zero mean, 〈 H$_{ij}$ 〉 = 0, and a variance decaying as a power law 〈| H$_{ij}$ | $^{2}$〉 ∼ ( b/ | i − j | ) 2 $^{α}$, where b and α are parame- ters. There are two prescriptions for the variance of the PBRM model: the so-called non-periodic, where the 1D sample is in a line geometry; and the peri- odic, where the sample is in a ring geometry. Field-theoretical considerations 4,30,31 and detailed numerical investiga- tions 4,32,33 have verified that the PBRM model undergoes a transition at α = 1 from localized states for α > 1 to delocalized states for α < 1. This transition shows all the key features of the Anderson MIT, including multifrac- tality of eigenfunctions and non-trivial spectral statistics at the critical point. Thus the PBRM model possesses a line of critical points b ∈ (0 , ∞ ). We set α = 1 in our study, i.e., we work with the PBRM model at criticality. The 3D Anderson model with diagonal disorder is de- scribed by the tight-binding Hamiltonian (TBH) where n ≡ ( n$_{x}$,n$_{y}$,n$_{z}$ ) labels all the N = L 3 sites of a cubic lattice with linear size L , while the second sum is taken over all nearest-neighbour pairs ( n , m ) on the lat- tice. The on-site potentials W$_{n}$ for 1 ≤ n$_{x}$,n$_{y}$,n$_{z}$ ≤ L are independent random variables. When W$_{n}$ are Gaus- sian distributed, with zero mean and variance W $^{2}$/ 12, the MIT at energy E ≃ 0 occurs for W = W$_{c}$ ≃ 21 . 3. See Refs. [17,34,35]. Then, for W < W$_{c}$ ( W > W$_{c}$ ) the system is in the metallic (insulating) regime. We set W = W$_{c}$ in our study. We open the isolated samples, defined above by the PBRM model and the 3D Anderson model, by attach- ing 2 M semi-infinite single channel leads. Each lead is described by the 1D semi-infinite TBH Using standard methods 36 one can write the scattering matrix ( S -matrix) in the form 11,37 where 1 is the 2 M × 2 M unit matrix, k = arccos( E/ 2) is the wave vector supported in the leads, and H eff is an effective non-hermitian Hamiltonian given by Here, W is an N × 2 M matrix that specifies the posi- tions of the attached leads to the sample. Its elements are equal to zero or ϵ , where ϵ is the coupling strength. Moreover, assuming that the wave vector k do not change significantly in the centre of the band, we set E = 0 and neglect the energy dependence of H eff and S . We start with the average of the S -matrix elements. It is known that Notice from Eqs. (1-2) that in the limit b → ∞ the PBRM model reproduces the Gaussian Orthogonal En- semble. Therefore, in that limit we expect the statistics of the scattering matrix, Eq. (5), to be determined by the Circular Orthogonal Ensemble (COE) which is the appropriate ensemble for systems with time reversal sym- metry. Thus, below, we provide the statistical results for the average S -matrix and the transport quantities to be analysed in the following sections, assuming the orthog- onal symmetry. In all cases, we also assume the absence of direct processes, i.e., 〈 S 〉 = 0. Within a scattering approach to the electronic trans- port, once the scattering matrix is known one can com- pute the dimensionless conductance T = Tr( tt $^{†}$) and its distribution w ( T ). For M = 1, i.e., considering two single-channels leads attached to the sample, w ( T ) is given by where 〈·〉 means ensemble average over the COE. For arbitrary M , the predictions for the average value of T and its variance are and respectively. For the derivation of the expressions above see for example Ref. [38]. Another transport quantity of interest is the shot noise power P = 〈 Tr( tt † − tt $^{†}$tt$^{†}$) 〉 , which as a function of M reads 39–41 〈| H$_{ij}$ | $^{2}$〉 = 1 2 1 + δ$_{ij}$ 1 + ( | i − j | /b ) 2 α , (1) 〈| H$_{ij}$ | $^{2}$〉 = 1 2 1 + δ$_{ij}$ 1 + [sin ( π | i − j | /L ) / ( πb/L )] 2 α , (2) H = ∑ n | n 〉 W$_{n}$ 〈 n | + ∑ ( n , m $_{)}$| n 〉〈 m | , (3) H lead = −∞ ∑ n =1 ( | n 〉〈 n + 1 | + | n + 1 〉〈 n | ) . (4) S ( E ) = ( r t ′ t r ′ ) = 1 − 2 i sin( k ) W $^{T}$( E − H $_{eff}$) − $^{1}$W , (5) H eff = H − e $^{ik}$W W T . (6) 〈 | S$_{aa}$ ′ | $^{2}$〉 COE = 1 + δ$_{aa}$ ′ 2 M + 1 , (7) w ( T ) COE = 1 $_{2}$√$_{T}$ , (8) w ( T ) COE = { 3 $_{2}$T , 0 < T < 1 3 2 ( T − $_{2}$√$_{T}$ − 1 ) , 1 < T < 2 . (9) while for M = 2, 〈 T 〉 COE = M 2 − M 2(2 M + 1) (10) var( T ) COE = M ( M + 1) 2 (2 M + 1) $^{2}$(2 M + 3) , (11) P COE = M ( M + 1) 2 2(2 M + 1)(2 M + 3) . (12) B. RMT predictions for the Circular Orthogonal Ensemble