Statistics of the two-point transmission at Anderson localization transitions

At Anderson critical points, the statistics of the two-point transmission $T_L$ for disordered samples of linear size $L$ is expected to be multifractal with the following properties [Janssen {\it et al} PRB 59, 15836 (1999)] : (i) the probability to have $T_L \sim 1/L^{\kappa}$ behaves as $L^{\Phi(\kappa)}$, where the multifractal spectrum $\Phi(\kappa)$ terminates at $\kappa=0$ as a consequence of the physical bound $T_L \leq 1$; (ii) the exponents $X(q)$ that govern the moments $\overline{T_L^q} \sim 1/L^{X(q)}$ become frozen above some threshold: $X(q \geq q_{sat}) = - \Phi(\kappa=0)$, i.e. all moments of order $q \geq q_{sat}$ are governed by the measure of the rare samples having a finite transmission ($\kappa=0$). In the present paper, we test numerically these predictions for the ensemble of $L \times L$ power-law random banded matrices, where the random hopping $H_{i,j}$ decays as a power-law $(b/| i-j |)^a$. This model is known to present an Anderson transition at $a=1$ between localized ($a>1$) and extended ($a<1$) states, with critical properties that depend continuously on the parameter $b$. Our numerical results for the multifractal spectra $\Phi_b(\kappa)$ for various $b$ are in agreement with the relation $\Phi(\kappa \geq 0) = 2 [ f(\alpha= d+ \frac{\kappa}{2}) -d ]$ in terms of the singularity spectrum $f(\alpha)$ of individual critical eigenfunctions, in particular the typical exponents are related via the relation $\kappa_{typ}(b)= 2 (\alpha_{typ}(b)-d)$. We also discuss the statistics of the two-point transmission in the delocalized phase and in the localized phase.