Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-01T02:54:29.402Z Has data issue: false hasContentIssue false
  • English
  • Français

On the sum of the non-negative Lyapunov exponents for some cocycles related to the Anderson model

Published online by Cambridge University Press:  06 October 2015

ILIA BINDER ,
MICHAEL GOLDSTEIN  and
MIRCEA VODA
ILIA BINDER
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada email [email protected], [email protected], [email protected]
MICHAEL GOLDSTEIN
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada email [email protected], [email protected], [email protected]
MIRCEA VODA
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada email [email protected], [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide an explicit lower bound for the the sum of the non-negative Lyapunov exponents for some cocycles related to the Anderson model. In particular, for the Anderson model on a strip of width $W$, the lower bound is proportional to $W^{-\unicode[STIX]{x1D716}}$, for any $\unicode[STIX]{x1D716}>0$. This bound is consistent with the fact that the lowest non-negative Lyapunov exponent is conjectured to have a lower bound proportional to $W^{-1}$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 2 , April 2017 , pp. 369 - 388
Copyright
© Cambridge University Press, 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aizenman, M. and Molchanov, S.. Localization at large disorder and at extreme energies: an elementary derivation. Comm. Math. Phys. 157(2) (1993), 245–278.CrossRefGoogle Scholar
Binder, I., Goldstein, M. and Voda, and M.. On fluctuations and localization length for the Anderson model on a strip. J. Spectr. Theory 5(1) (2015), 193–225.CrossRefGoogle Scholar
Bougerol, P. and Lacroix, J.. Products of Random Matrices with Applications to Schrödinger Operators (Progress in Probability and Statistics, 8) . Birkhäuser Boston Inc., Boston, MA, 1985.CrossRefGoogle Scholar
Bourgain, J.. A lower bound for the Lyapounov exponents of the random Schrödinger operator on a strip. J. Stat. Phys. 153(1) (2013), 1–9.CrossRefGoogle Scholar
Combes, J.-M., Germinet, F. and Klein, A.. Generalized eigenvalue-counting estimates for the Anderson model. J. Stat. Phys. 135(2) (2009), 201–216.CrossRefGoogle Scholar
Carmona, R. and Lacroix, J.. Spectral Theory of Random Schrödinger Operators (Probability and its Applications) . Birkhäuser Boston Inc., Boston, MA, 1990.CrossRefGoogle Scholar
Craig, W. and Simon, B.. Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices. Comm. Math. Phys. 90(2) (1983), 207–218.CrossRefGoogle Scholar
Gol’dsheĭd, I. Ya. and Margulis, G. A.. The condition of simplicity for the spectrum of Lyapunov exponents. Dokl. Akad. Nauk SSSR 293(2) (1987), 297–301.Google Scholar
Horn, R. A. and Johnson, C. R.. Matrix Analysis. Cambridge University Press, Cambridge, 1985.CrossRefGoogle Scholar
Klein, A., Lacroix, J. and Speis, A.. Localization for the Anderson model on a strip with singular potentials. J. Funct. Anal. 94(1) (1990), 135–155.CrossRefGoogle Scholar
Kotani, S. and Simon, B.. Stochastic Schrödinger operators and Jacobi matrices on the strip. Comm. Math. Phys. 119(3) (1988), 403–429.CrossRefGoogle Scholar
Petrov, V. V.. Limit Theorems of Probability Theory: Sequences of Independent Random Variables (Oxford Studies in Probability, 4) . The Clarendon Press, Oxford University Press, New York, 1995.Google Scholar
Schenker, J.. Eigenvector localization for random band matrices with power law band width. Comm. Math. Phys. 290(3) (2009), 1065–1097.CrossRefGoogle Scholar
Tornheim, L.. The Sylvester–Franke theorem. Amer. Math. Monthly 59 (1952), 389–391.CrossRefGoogle Scholar
Zhang, F. (Ed). The Schur Complement and its Applications (Numerical Methods and Algorithms, 4) . Springer, New York, 2005.CrossRefGoogle Scholar