I present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products $\Pi_n=M_nM_{n-1}\cdots M_1$, where $M_i$'s are i.i.d.. Following Tutubalin [Theor. Probab. Appl. {\bf 10}, 15 (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 group $\mathrm{SL}(2,\mathbb{R})$ where explicit calculations are possible. For concreteness, I study in detail transfer matrix products for the one-dimensional Schrodinger equation where the random potential is a Levy noise (derivative of a Levy process). In this case, I obtain a general formula for the variance of $\ln||\Pi_n||$ and for the variance of $\ln|\psi(x)|$, where $\psi(x)$ is 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.

中文翻译：

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随机矩阵与广义李雅普诺夫指数乘积的波动

我提出了一个通用框架，允许对随机矩阵乘积 $\Pi_n=M_nM_{n-1}\cdots M_1$ 的矩生成函数进行显式计算，其中 $M_i$ 是 iid。遵循 Tutubalin [理论。可能。应用程序 {\bf 10}, 15 (1965)]，生成函数的计算被简化为找到与该组的一系列表示相关联的某个传递算子的最大特征值。形式主义通过考虑来自群 $\mathrm{SL}(2,\mathbb{R})$ 的随机矩阵的乘积来说明，其中可以进行显式计算。为具体起见，我详细研究了一维薛定谔方程的传递矩阵乘积，其中随机势是 Levy 噪声（Levy 过程的导数）。在这种情况下，我得到了 $\ln||\Pi_n||$ 的方差和 $\ln|\psi(x)|$ 的方差的一般公式，其中 $\psi(x)$ 是波函数，表示为涉及矩阵乘积不变密度的傅立叶变换的单个积分。最后我讨论了随机矩阵乘积（接近恒等式的矩阵）的连续极限。特别是，我研究了一个简单的案例，在这个案例中，提供广义李雅普诺夫指数的谱问题可以被精确地解决。