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Large Deviations for Products of Random Two Dimensional Matrices

  • Pedro Duarte
  • Silvius KleinEmail author
Article
  • 34 Downloads

Abstract

We establish large deviation type estimates for i.i.d. products of two dimensional random matrices with finitely supported probability distribution. The estimates are stable under perturbations and require no irreducibility assumptions. In consequence, we obtain a uniform local modulus of continuity for the corresponding Lyapunov exponent regarded as a function of the support of the distribution. This in turn has consequences on the modulus of continuity of the integrated density of states and on the localization properties of random Jacobi operators.

Notes

Acknowledgement

Pedro Duarte was supported by Fundação para a Ciência e a Tecnologia, under the projects: UID/MAT/04561/2013 and PTDC/MAT-PUR/29126/2017. Silvius Klein has been supported in part by the CNPq research Grant 306369/2017-6 (Brazil) and by a research productivity grant from his institution (PUC-Rio). He would also like to acknowledge the support of the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, where the work on this project first started, during the PEPW04 workshop in 2015. Both authors are grateful to the anonymous referees for their diligent reading of the manuscript and their useful suggestions for improvement.

References

  1. 1.
    Baraviera, A., Duarte, P.: Approximating Lyapunov exponents and stationary measures. J. Dyn. Differ. Equ. 31(1), 25–48 (2019)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bocker-Neto, C., Viana, M.: Continuity of Lyapunov exponents for random two-dimensional matrices. Ergod. Theory Dyn. Syst. 37(5), 1413–1442 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bougerol, P.: Théorèmes limite pour les systèmes linéaires à coefficients markoviens. Probab. Theory Relat. Fields 78(2), 193–221 (1988)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bourgain, J., Goldstein, M.: On nonperturbative localization with quasi-periodic potential. Ann. Math. (2) 152(3), 835–879 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bourgain, J., Schlag, W.: Anderson localization for Schrödinger operators on \( Z\) with strongly mixing potentials. Commun. Math. Phys. 215(1), 143–175 (2000)Google Scholar
  6. 6.
    Bucaj, V., Damanik, D., Fillman, J., Gerbuz, V., VandenBoom, T., Wang, F., Zhang, Z.: Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent, preprint (2017)Google Scholar
  7. 7.
    Chapman, J., Stolz, G.: Localization for random block operators related to the XY spin chain. Ann. Henri Poincaré 16(2), 405–435 (2015)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Damanik, D.: Schrödinger operators with dynamically defined potentials. Ergod. Theory Dyn. Syst. 37(6), 1681–1764 (2017)CrossRefGoogle Scholar
  9. 9.
    Duarte, P., Klein, S.: Lyapunov exponents of linear cocycles: continuity via large deviations. In: Atlantis Studies in Dynamical Systems, vol. 3. Atlantis Press, Series Editors: Broer, Henk, Hasselblatt, Boris, Paris (2016) Google Scholar
  10. 10.
    Duarte, P., Klein, S.: Continuity of the Lyapunov exponents of linear cocycles, Publicações Matemáticas, \(31^\circ \) Colóquio Brasileiro de Matemática, IMPA. https://impa.br/wp-content/uploads/2017/08/31CBM_02.pdf (2017)
  11. 11.
    Duarte, P., Klein, S., Santos, M.: A random cocycle with non Hölder Lyapunov exponent. Discret. Contin. Dyn. Syst. A 39(8), 4841–4861 (2019)CrossRefGoogle Scholar
  12. 12.
    Furstenberg, H., Kifer, Y.: Random matrix products and measures on projective spaces. Isr. J. Math. 46(1–2), 12–32 (1983)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Goldstein, M., Schlag, W.: Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. Math. (2) 154(1), 155–203 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Han, R., Lemm, M., Schlag, W.: Effective multi-scale approach to the Schrödinger cocycle over a skew-shift base. Ergod. Theory Dyn. Syst. (2019).  https://doi.org/10.1017/etds.2019.19
  15. 15.
    Jitomirskaya, S.: Metal-insulator transition for the almost Mathieu operator. Ann. Math. (2) 150(3), 1159–1175 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jitomirskaya, S., Zhu, X.: Large deviations of the Lyapunov exponent and localization for the 1D Anderson model. Commun. Math. Phys. 370(1), 311–324 (2019)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kato, T.: Perturbation theory for linear operators. In: Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 editionGoogle Scholar
  18. 18.
    Kloeckner, B.R.: Effective perturbation theory for simple isolated eigenvalues of linear operators. J. Oper. Theory 81(1), 175–194 (2019)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Le Page, É.: Théorèmes limites pour les produits de matrices aléatoires. In: Heyer, H. (ed.) Probability Measures on Groups (Oberwolfach, 1981). Lecture Notes in Mathematics, vol. 928, pp. 258–303. Springer, Berlin (1982)CrossRefGoogle Scholar
  20. 20.
    Le Page, É.: Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications. Ann. Inst. H. Poincaré Probab. Stat. 25(2), 109–142 (1989)zbMATHGoogle Scholar
  21. 21.
    Ledrappier, F.: Quelques propriétés des exposants caractéristiques. In: Hennequin, P.L. (ed.) École d’été de probabilités de Saint-Flour, XII–1982. Lecture Notes in Mathematics, vol. 1097, pp. 305–396. Springer, Berlin (1984)CrossRefGoogle Scholar
  22. 22.
    Malheiro, E.C., Viana, M.: Lyapunov exponents of linear cocycles over Markov shifts. Stoch. Dyn. 15(3), 1550020 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Stolz, G.: An introduction to the mathematics of Anderson localization. In: Sims, R., Ueltschi, D. (eds.) Entropy and the Quantum II: Contemporary Mathematics, vol. 552, pp. 71–108. American Mathematical Society, Providence (2011)CrossRefGoogle Scholar
  24. 24.
    El Hadji, Y.T., Viana, M.: Moduli of continuity for Lyapunov exponents of random GL(2) cocycles, preprint (2018)Google Scholar
  25. 25.
    Tao, T.: Topics in Random Matrix Theory. Graduate Studies in Mathematics, vol. 132. American Mathematical Society, Providence (2012)CrossRefGoogle Scholar
  26. 26.
    Viana, M.: Lectures on Lyapunov Exponents. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2014)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Matemática and CMAFcIO, Faculdade de CiênciasUniversidade de LisboaLisbonPortugal
  2. 2.Departamento de MatemáticaPontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)Rio de JaneiroBrazil

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