大随机矩阵的定量富斯滕贝格理论
Quantitative Furstenberg Theory for Large Random Matrices
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中文总结 AI 辅助
该研究针对具有GOE势块的块安德森模型的2W×2W转移矩阵，定量发展李雅普诺夫指数理论，证明其极限定理、指数间隙下界及局域化长度上界，采用富斯滕贝格公式与马利亚万微积分等方法完成证明。
中文摘要 AI 辅助
我们考虑与具有GOE势块的块安德森模型相关的2W×2W转移矩阵。我们以两种方式使经典李雅普诺夫指数理论定量化：首先，证明了最大李雅普诺夫指数的定量极限定理；其次，证明任意两个李雅普诺夫指数之间的间隙至少为c/W。作为推论，这意味着该一维块安德森模型的局域化长度最多为CW²。证明使用了李雅普诺夫指数的富斯滕贝格型公式，并在辛群中采用马利亚万微积分风格的论证，以证明足够多转移矩阵的乘积具有足够光滑的密度，后者的主要技术输入是结构化随机矩阵的最小奇异值估计。
英文摘要
We consider the $2W\times 2W$ transfer matrices associated to the block Anderson model with GOE potential blocks. We make the classical Lyapunov exponent theory quantitative in two ways. First, we prove a quantitative limit theorem for the top Lyapunov exponent. Second, we prove every gap between Lyapunov exponents is at least $c/W$. As a corollary, this implies the localization length of this $1$d block Anderson model is at most $CW^2$. The proof uses Furstenberg type formulas for the Lyapunov exponents, and Malliavin calculus style arguments in the symplectic group to show the product of sufficiently many transfer matrices has a sufficiently smooth density. The main technical input for the latter is a least singular value estimate for a structured random matrix.
