Fluctuations of the Product of Random Matrices and Generalized Lyapunov Exponent
Fluctuations of the Product of Random Matrices and Generalized Lyapunov Exponent
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随机矩阵与广义李雅普诺夫指数乘积的涨落
DOI：
10.1007/s10955-020-02617-w
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发表时间：
2019
影响因子：
1.6
通讯作者：
C. Texier
中科院分区：
文献类型：
--
作者：
C. Texier
我提出了一个允许对随机矩阵积的矩生成函数进行显式计算的一般框架，其中为i.i.d。根据Tutubalin (Theor Probab Appl 10(1):15 - 27,1965)，生成函数的计算简化为寻找与群的一组表示相关的某个传递算子的最大特征值。通过考虑可能进行显式计算的群中的随机矩阵的乘积来说明这种形式。具体而言，我详细研究了一维Schrödinger方程的传递矩阵积，其中随机势是一个l<s:1>杂讯（l<s:1>杂讯过程的导数）。在这种情况下，我得到了方差和方差的一般公式，其中的方差是波函数，用一个包含矩阵乘积不变密度的傅里叶变换的积分表示。最后讨论了随机矩阵积（接近恒等矩阵）的连续极限。特别地，我研究了一个简单的例子，其中提供广义李雅普诺夫指数的谱问题可以精确地解决。
I present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products, where’s are i.i.d. Following Tutubalin (Theor Probab Appl 10(1):15–27, 1965), the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group. The formalism is illustrated by considering products of random matrices from the groupwhere explicit calculations are possible. For concreteness, I study in detail transfer matrix products for the one-dimensional Schrödinger equation where the random potential is a Lévy noise (derivative of a Lévy process). In this case, I obtain a general formula for the variance ofand for the variance of, whereis the wavefunction, in terms of a single integral involving the Fourier transform of the invariant density of the matrix product. Finally I discuss the continuum limit of random matrix products (matrices close to the identity). In particular, I investigate a simple case where the spectral problem providing the generalized Lyapunov exponent can be solved exactly.
影响因子：
8.6
作者：
ABRAHAMS, E;ANDERSON, PW;RAMAKRISHNAN, TV
通讯作者：
RAMAKRISHNAN, TV