Large Deviations for Products of Random Two Dimensional Matrices
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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.
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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