Constructive fractional-moment criteria for localization in random operators

Physica A: Statistical Mechanics and its Applications

Volumn 279, Issue 1, 2000, Pages 369-377

  Aizenman, Michael ^a   Schenker, Jeffrey H ^a   Friedrich, Roland M ^b   Hundertmark, Dirk ^c  

^a PRINCETON UNIVERSITY   (United States)

^b ETH ZURICH   (Switzerland)

^c CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ELECTRIC CONDUCTANCE; GREEN'S FUNCTION; HALL EFFECT; MATHEMATICAL OPERATORS; QUANTUM THEORY; RANDOM PROCESSES;

FERMI GASES; FRACTIONAL MOMENTS; HALL CONDUCTANCE; RANDOM OPERATORS;

STATISTICAL MECHANICS;

EID: 0033727833     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0378-4371(00)00012-1     Document Type: Article

References (29)

1. E. Abraham, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42 (1979) 673.
2. A.D. Mirlin, Y.V. Fyodorov, Localization transition in the Anderson model on the Bethe lattice: spontaneous symmetry breaking and correlation function, Nucl. Phys. B 366 (1991) 507.
3. M. Aizenman, J.H. Schenker, R.M. Friedrich, D. Hundertmark, Finite-volume criteria for Anderson localization, preprint, 1999, http://xxx.lanl.gov/abs/math-ph/9910022.
4. M. Aizenman, S. Molchanov, Localization at large disorder and at extreme energies: an elementary derivation, Commun. Math. Phys. 157 (1999) 245.
5. B. Simon, T. Wolff, Singular continuous spectra under rank one perturbation and localization for random Hamiltonians, Commun. Pure Appl. Math. 39 (1996) 75.
6. M. Aizenman, Localization at weak disorder: some elementary bounds, Rev. Math. Phys. 6 (1994) 1163.
7. R. del Rio, S. Jitomirskaya, Y. Last, B. Simon, Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69 (1996) 153.
8. N. Minami, Local fluctuation of the spectrum of a multidimensional Anderson tight binding model, Commun. Math. Phys. 177 (1996) 709.
9. M. Aizenman, G.M. Graf, Localization bounds for an electron gas, J. Phys. A 31 (1998) 6783.
10. J.E. Avron, R. Seiler, B. Simon, Charge deficiency, charge transport and comparison of dimensions, Commun. Math. Phys. 159 (1994) 399.
11. J. Bellissard, A. van Eist, H. Schulz-Baldes, The noncommutative geometry of the quantum Hall effect, J. Math. Phys. 35 (1994) 5373.
12. J. Fröhlich, T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Commun. Math. Phys. 88 (1993) 151.
13. D. Damanik, P. Stollmann, Multi-scale analysis implies strong dynamical localization, preprint, 1999 http://xxx.lanl.gov/abs/math-ph/9912002.
14. R.L. Dobrushin, S.B. Shlosman, Completely analytical interactions: constructive description, J. Statist. Phys. 46 (5-6) (1987) 983-1014.
15. J.M. Hammersley, Percolation processes II. The connective constant, Proc. Camb. Philos. Soc. 53 (1957) 642.
16. M. Aizenman, C.M. Newman, Tree graph inequalities and critical behavior in percolation models, J. Statist. Phys. 36 (1984) 107.
17. B. Simon, Correlation inequalities and the decay of correlations in ferromagnets, Commun. Math. Phys. 77 (2) (1980) 111.
18. E.H. Lieb, A refinement of Simon's correlation inequality, Commun. Math. Phys. 77 (2) (1980) 127.
19. M. Aizenman, B. Simon, Local Ward identities and the decay of correlations in ferromagnets, Commun. Math. Phys. 77 (2) (1980) 137.
20. M. Aizenman, R. Holley, Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin Shlosman regime, in: Percolation theory and ergodic theory of infinite particle systems (Minneapolis, Minn., 1984-1985), Vol 1, Springer, New York, 1987.
21. C. Maes, S.B. Shlosman, Ergodicity of probabilistic cellular automata: a constructive criterion, Commun. Math. Phys. 135 (2) (1991) 233.
22. H. Kunz, B. Souillard, Sur le spectre des operateurs aux différences finies aléatoires, Commun. Math. Phys. 78 (201) (1980).
23. A. Klein, Absolutely continuous spectrum in the Ajiderson model on the Bethe lattice, Math. Res. Lett. 1 (1993) 399.
24. B. Simon, Lifschitz tails for the Ajiderson model, J. Stat. Phys. 38 (1985) 65.
25. J.M. Barbaroux, J.M. Combes, P.D. Hislop, Localization near band edges for random Schrödinger operators, Helv. Phys. Acta 70 (1997) 16. (Papers honouring the 60th birthday of Klaus Hepp and of Walter Hunziker, Part II Zürich, 1995).
26. A. Figotin, A. Klein, Localization of electromagnetic and acoustic waves in random media, lattice models, J. Stat. Phys. 76 (1994) 985.
27. W. Kirsch, P. Stollmann, G. Stolz, Localization for random perturbations of periodic Schrödinger operators. Rand. Oper. Stochastics Equations (6) 1998.
28. F. Klopp, Internal Lifshits tails for random perturbations of periodic Schrödinger operators., Duke Math. J. 98 (1999) 335.
29. P. Stollmann, Lifshitz asymptotics via liner coupling of disorder, preprint, 1999.