Quantum Conduction on a Cayley Tree

Quantum resistance of a Cayley tree composed of one-dimensional random scatterers is considered. The strength of a scatterers is characterized by its typical resistance ${\rho }_0$. It is shown that a metal-insulator transition occurs at some critical scattering strength ${\rho }_{0c}$, which depends on the coordination number of the tree. The resistance of an infinite tree diverges for ${\rho }_0>{\rho }_{0c}$ and saturates at some finite value for ${\rho }_0<{\rho }_{0c}$.