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Large Deviations for Products of Random Two Dimensional Matrices

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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.

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Notes

  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.

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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.

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Authors and Affiliations

  1. Departamento de Matemática and CMAFcIO, Faculdade de Ciências, Universidade de Lisboa, Lisbon, Portugal

    Pedro Duarte

  2. Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil

    Silvius Klein

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  1. Pedro Duarte
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  2. Silvius Klein
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Correspondence to Silvius Klein.

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Communicated by W. Schlag

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Duarte, P., Klein, S. Large Deviations for Products of Random Two Dimensional Matrices. Commun. Math. Phys. 375, 2191–2257 (2020). https://doi.org/10.1007/s00220-019-03586-2

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