Abstract
We consider the usual one-dimensional tight-binding Anderson model with the random potential taking only two values, 0 andλ, with probabilityp and 1−p, 0<p<1. We show that the Liapunov exponentγ λ (E), E ε R. diverges asλ → ∞ uniformly in the energyE. Using a result of Carmona, Klein, and Martinelli, this proves that forλ large enough, the integrated density of states is singular continuous. We also compute explicitly the exact asymptotics for a dense set of energies and we compare the results with numerical simulations.
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References
H. Schmidt,Phys. Rev. 105:425 (1957).
K. Ishii,Suppl. Progr. Theor. Phys. 53:77 (1973).
P. Bougerol and J. Lacroix,Products of Random Matrices with Application to Schrödinger Operators (Birkhauser, Boston, 1985).
R. Carmona, A. Klein, and F. Martinelli,Comm. Math. Phys., to appear.
B. Simon and M. Taylor,Comm. Math. Phys. 101:1 (1985).
D. J. Thouless,J. Phys. C 5:77 (1972).
F. Wegner,Z. Phys. B 44:9 (1981).
E. Le Page, Empirical distribution of the eigenvalues of a Jacobi matrix, Probability measures on groups,VII Springer Lecture Notes Series 1064 (1983), p. 309.
B. Halperin,Adv. Chem. Phys. 13:123 (1967).
Th. M. Nieuwenhuizen and J. M. Luck,J. Stat. Phys. Vol.41:No. 5/6, 745 (1985).
B. Derrida and E. Gardner,J. Phys. (Paris) 45:1283 (1984).
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Martinelli, F., Micheli, L. On the large-coupling-constant behavior of the Liapunov exponent in a binary alloy. J Stat Phys 48, 1–18 (1987). https://doi.org/10.1007/BF01010397
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DOI: https://doi.org/10.1007/BF01010397