Abstract
We propose a new probabilistic approach to the analysis of decay of the Green's functions and the eigenfunctions of the Anderson Hamiltonians on count- able graphs. Our method is close in spirit to the Frac- tional Moment Method, but we show how the use of the fractional moments can be avoided, so that exponen- tial decay of the Green's functions can be established in some models where the fractional moments diverge, due to low regularity of the random potential. We elucidate the exceptional role of the Holder continuity condition, usual in the FMM, in terms of Cramer's condition in the large deviations problem for a suitably constructed rigorous path expansion.
Highlights
In the framework of the Fractional Moment Method (FMM), this comes as a matter-of-fact constatation; since the method stops working for less regular random potentials, the analysis of the role of the Holder continuity stops there
We will allow for a lower regularity of the probability distribution of the random potential, loosing the benefit of Cramer’s condition in the large deviations analysis
One can trade higher regularity of the random potential for higher rate of growth of balls. It is clear from Eqn (4.7) that, in the graphs with exponential growth of balls, one needs exponential bounds on the probabilities of large deviations to overcome the exponentially large factor S(L), so our technique, just as the FMM, requires Holder continuity of the marginal distribution for the proof of dynamical localization in this class of models
Summary
Decoupling techniques in the approach, based on the complete decoupling of the fraclocalization theory tional moments of the Green’s functions in small subsys-. Recall that Martinelli and Scoppola [26] proved the absence of the a.c. spectrum on a lattice under the assumption of fast decay of the Green’s functions Their argument, based on the Chebyshev inequality in the energy-disorder product space, has been further developed and used, e.g., by Bourgain and Kenig [9], where it is a part of an elaborate, all-in-on scaling procedure, and, in a more distinctly encapsulated form, by Elgart et al [15]. (3) In Section 3, we employ the “slim wormhole” expansions and standard methods of the large deviations analysis to prove exponential decay of Green’s functions for Anderson Hamiltonians on various types of locally finite graphs. (5) The large deviations estimates used in the main text are proven in Appendix
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