Anderson localization on the Cayley tree : multifractal statistics of the transmission at criticality and off criticality

In contrast to finite dimensions where disordered systems display multifractal statistics only at criticality, the tree geometry induces multifractal statistics for disordered systems also off criticality. For the Anderson tight-binding localization model defined on a tree of branching ratio K=2 with $N$ generations, we consider the Miller-Derrida scattering geometry [J. Stat. Phys. 75, 357 (1994)], where an incoming wire is attached to the root of the tree, and where $K^N$ outcoming wires are attached to the leaves of the tree. In terms of the $K^N$ transmission amplitudes $t_j$, the total Landauer transmission is $T\equiv {\sum }_j\mathrm{\mid }{t}_j{\mathrm{\mid }}^2$, so that each channel $j$ is characterized by the weight $w_j=\mathrm{\mid }{t}_j{\mathrm{\mid }}^2\mathrm{/}T$. We numerically measure the typical multifractal singularity spectrum $f(\alpha )$ of these weights as a function of the disorder strength $W$ and we obtain the following conclusions for its left-termination point ${\alpha }_{+}(W)$. In the delocalized phase $W<W_c$, ${\alpha }_{+}(W)$ is strictly positive ${\alpha }_{+}(W)>0$ and is associated with a moment index $q_{+}(W)>1$. At criticality, it vanishes ${\alpha }_{+}(W_c)=0$ and is associated with the moment index $q_{+}(W_c)=1$. In the localized phase $W>W_c$, ${\alpha }_{+}(W)=0$ is associated with some moment index $q_{+}(W)<1$. We discuss the similarities with the exact results concerning the multifractal properties of the Directed Polymer on the Cayley tree.