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The oscillation theorem discussed in Sect. 2 now gives (17). 46 S. Jitomirskaya, H. Schulz-Baldes, G. Stolz 6. Lower Bound on Dynamics The deterministic part of the argument presented in this section follows [DT]. Let us return to the simplified notation from Sect. 2 and write ω instead of (ω, l) since based on the results of Sect. 5 the value of l will not influence the considerations. Let us begin with some preliminaries and introduce the Green’s function Gzω (n) = n 1 0 . Hω − z Note that −tω (n + 1) Gzω (n + 1) + (vω (n) − Ec − z) Gzω (n) − tω (n) Gzω (n − 1) = δn,0 .

If the sequence {λm } also satisfies the boundary condition (8) and the λm ’s are not all zero then (i) 0 < λm < s, for all m. (ii) λm tends monotonically to 0 for m large enough. (iii) m V (λm ) = 0 and m λm < ∞. 54 B. Durhuus, T. Jonsson, R. Nest V’(x) t r s w x Fig. 1. A graph of the derivative of a generic potential V which satisfies our assumptions Dropping the assumption of rotational symmetry we have the following generalization of (i) and (iii), which, apart from being of some independent interest, we will use in Sect.

The latter implies for all configurations a weaker lower bound in (16) of the form 1 − q1 . One can construct configurations ω with slower transport than in (16). Therefore – seemingly paradoxically – typical random configurations do not lead the slowest possible transport for this model. Finally it is worth mentioning a large deviation result here. The IDS and Lyapunov exponent are both averaged quantities describing the behavior at the infinite volume limit. t. P) the following holds for the finite (but sufficiently large) size Hamiltonian Hω,N found by restricting Hω to 2 ({0, .

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Communications in Mathematical Physics - Volume 233 by M. Aizenman (Chief Editor)

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