Skip to main content

Large Deviations for Products of Random Two Dimensional Matrices


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.

This is a preview of subscription content, access via your institution.

Fig. 1


  1. Irreducibility refers to the non existence of proper subspaces invariant under the closed semigroup \(T_\mu \) generated by the support of the measure \(\mu \). There are different versions of this notion.

  2. Contractivity refers to the existence in \(T_\mu \) of matrices with arbitrarily large gaps between consecutive singular values.

  3. A more detailed review of such results—including another new quantitative statement—follows.

  4. We note that the case of an absolutely continuous probability distribution was already studied by E. Le Page.

  5. We refer to the constants c and \({\bar{n}}\) as the LDT parameters of A. They depend on A, and in general they may blow up as A is perturbed.

  6. Thus we obtain another proof of the result of C. Bocker and M. Viana on the continuity of the LE on the whole space of cocycles, in the finite support setting.

  7. This method is based on ideas introduced by Goldstein and Schlag [13] in their study of quasi-periodic Schrödinger operators.

  8. We later prove (see Lemma 3.1) that when , this is equivalent to the concept of quasi-irreducibility introduced in [3] (see Definition 2.7).

  9. This claim, in a more general setting, is also proven in [3, Proposition 2.8], using ingredients in the proof of the Oseledets multiplicative ergodic theorem given by Ledrappier [21].

  10. With this definition, ,  for all \(\delta >0\).

  11. The case where \(\rho (B)= \rho _+(B)>0\) reduces to the previous one applied to the inverse cocycle \(B^{-1}\) and will not be addressed here.

  12. In [10] the uniform LDT is exponential, and as a result, the modulus of continuity of the LE and that of the Oseledets decomposition are Hölder. As alluded to at the end of Section 3.2 of [10] (or using our more general ACT in [9]), a sub-exponential uniform LDT leads to a weak-Hölder modulus of continuity.


  1. Baraviera, A., Duarte, P.: Approximating Lyapunov exponents and stationary measures. J. Dyn. Differ. Equ. 31(1), 25–48 (2019)

    MathSciNet  MATH  Google Scholar 

  2. Bocker-Neto, C., Viana, M.: Continuity of Lyapunov exponents for random two-dimensional matrices. Ergod. Theory Dyn. Syst. 37(5), 1413–1442 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Bougerol, P.: Théorèmes limite pour les systèmes linéaires à coefficients markoviens. Probab. Theory Relat. Fields 78(2), 193–221 (1988)

    MATH  Google Scholar 

  4. Bourgain, J., Goldstein, M.: On nonperturbative localization with quasi-periodic potential. Ann. Math. (2) 152(3), 835–879 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Bourgain, J., Schlag, W.: Anderson localization for Schrödinger operators on \( Z\) with strongly mixing potentials. Commun. Math. Phys. 215(1), 143–175 (2000)

    ADS  MATH  Google Scholar 

  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)

  7. Chapman, J., Stolz, G.: Localization for random block operators related to the XY spin chain. Ann. Henri Poincaré 16(2), 405–435 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  8. Damanik, D.: Schrödinger operators with dynamically defined potentials. Ergod. Theory Dyn. Syst. 37(6), 1681–1764 (2017)

    MATH  Google Scholar 

  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)

  10. Duarte, P., Klein, S.: Continuity of the Lyapunov exponents of linear cocycles, Publicações Matemáticas, \(31^\circ \) Colóquio Brasileiro de Matemática, IMPA. (2017)

  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)

    MATH  Google Scholar 

  12. Furstenberg, H., Kifer, Y.: Random matrix products and measures on projective spaces. Isr. J. Math. 46(1–2), 12–32 (1983)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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).

  15. Jitomirskaya, S.: Metal-insulator transition for the almost Mathieu operator. Ann. Math. (2) 150(3), 1159–1175 (1999)

    MathSciNet  MATH  Google Scholar 

  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)

    ADS  MathSciNet  MATH  Google Scholar 

  17. Kato, T.: Perturbation theory for linear operators. In: Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 edition

  18. Kloeckner, B.R.: Effective perturbation theory for simple isolated eigenvalues of linear operators. J. Oper. Theory 81(1), 175–194 (2019)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Google Scholar 

  22. Malheiro, E.C., Viana, M.: Lyapunov exponents of linear cocycles over Markov shifts. Stoch. Dyn. 15(3), 1550020 (2015)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  24. El Hadji, Y.T., Viana, M.: Moduli of continuity for Lyapunov exponents of random GL(2) cocycles, preprint (2018)

  25. Tao, T.: Topics in Random Matrix Theory. Graduate Studies in Mathematics, vol. 132. American Mathematical Society, Providence (2012)

    MATH  Google Scholar 

  26. Viana, M.: Lectures on Lyapunov Exponents. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Silvius Klein.

Additional information

Communicated by W. Schlag

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duarte, P., Klein, S. Large Deviations for Products of Random Two Dimensional Matrices. Commun. Math. Phys. 375, 2191–2257 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: