Anderson Transition and Generalized Lyapunov Exponents (comment on Comment

The generalized Lyapunov exponents describe the growth of the second moments for a particular solution of the quasi-1D Schroedinger equation with initial conditions on the left end. Their possible application in the Anderson transition theory became recently a subject for controversy in the literature. The approach to the problem of the second moments advanced by Markos et al (cond-mat/0402068) is shown to be trivially incorrect. The difference of approaches by Kuzovkov et al (cond-mat/0212036, 0501446) and the present author (cond-mat/0504557, 0512708) is discussed. Recently Markos et al have published a comment [1] on the paper by Kuzovkov et al [2] where a growth of the second moments for a particular solution of the quasi-1D Schroedinger equation was related with the problem of Anderson localization. It was stated in [2] that the Anderson transition is of the first order and exists not only for space dimensions d > 2 [3] but also in the 2D case. These statements look wild and the authors of [1] are right in not believing them. They are also right in statement that the growth of the second moment of wave function does not mean the growth of its typical value, which is governed by the average logarithm. However, Markos et al [1] go further and claim that analysis of the second moments presented by Kuzovkov et al [2, 3] is qualitatively incorrect and cannot provide any evidence for the metallic phase. These conclusions contradict the recent papers by the present author [4, 5] and are shown below to be trivially incorrect. The difference of approaches suggested in [2, 3] and [4, 5] is also discussed. Consider the 2D Anderson model described by the discrete Schroedinger equation and interprete it as a recurrence relation in the variable n, which we accept as a longitudinal coordinate. Initial conditions are assumed to be fixed on the left end of the system, while the periodic boundary conditions are accepted in the transverse direction, ψ n,m+L = ψ n,m. Cite energies V n,m are considered as uncorrelated random quantities with the first two moments The growth of the second moments for this problem can be studied using the old idea by Thouless [6] based on the observation that variables ψ n,m are statistically independent of 1