  We consider disordered tight-binding models which Green's functions obey the
self-consistent cavity equations . Based on these equations and the replica
representation, we derive an analytical expression for the fractal dimension
D_{1} that distinguishes between the extended ergodic, D_{1}=1, and extended
non-ergodic (multifractal), 0<D_{1}<1 states. The latter corresponds to the
solution with broken replica symmetry, while the former corresponds to the
replica-symmetric solution. We prove the existence of the extended non-ergodic
phase in a broad range of disorder strength and energy as well as existence of
transition between the two extended phases. The results are applied to the
systems with local tree structure (Bethe lattices) and to the systems with
infinite connectivity (Rosenzweig-Poter random matrix theory). We obtain the
phase diagram in the disorder-energy plain for the Bethe lattice and identify
two insulating phases classified by the (one-step) replica symmetry breaking
parameter. Finally we express the line of the Anderson localization transition,
the stability limit of the non-ergodic extended phase and the line of the first
order transitions between the two extended phases in terms of the Lyapunov
exponents.

Disordered Systems and Neural Networks;Statistical Mechanics