论文标题
来自多元基质不平等的Lyapunov指数上的定量下限
Quantitative lower bounds on the Lyapunov exponent from multivariate matrix inequalities
论文作者
论文摘要
Lyapunov指数表征了长基质产物的渐近行为。认识到Lyapunov指数严格为积极的方案是在许多应用中相关的基本挑战。在这项工作中，我们通过根据矩阵总和在lyapunov指数上得出定量下限来为该任务建立一个新颖的工具，该矩阵总和可以在ergodic情况下有效地计算。我们的方法结合了矩阵分析的两个深层结果---金汤普森不平等和雪崩原则的$ n $ -matrix扩展。我们将这些界限应用于SchrödingerCocycles的Lyapunov指数，该指数具有一定的聚合物类型和任意相关结构的千古势。我们还得出了与对角线矩阵附近的lyapunov指数相关的定量稳定性结果，并绑定了几乎指示矩阵的结合。
The Lyapunov exponent characterizes the asymptotic behavior of long matrix products. Recognizing scenarios where the Lyapunov exponent is strictly positive is a fundamental challenge that is relevant in many applications. In this work we establish a novel tool for this task by deriving a quantitative lower bound on the Lyapunov exponent in terms of a matrix sum which is efficiently computable in ergodic situations. Our approach combines two deep results from matrix analysis --- the $n$-matrix extension of the Golden-Thompson inequality and the Avalanche-Principle. We apply these bounds to the Lyapunov exponents of Schrödinger cocycles with certain ergodic potentials of polymer type and arbitrary correlation structure. We also derive related quantitative stability results for the Lyapunov exponent near aligned diagonal matrices and a bound for almost-commuting matrices.
