论文标题
Lyapunov指数在2D矩阵Langevin动力学中的大偏差，并应用于一维安德森本地化模型
Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models
论文作者
论文摘要
对于2D矩阵Langevin动力学，对应于大约2美元的2 $ 2 $随机矩阵的连续时间限制，有限的lyapunov指数可以写成相关的Riccati过程的加性功能，该功能提交给了无限次周期环上的某些Langevin动态。因此，可以从两种观点分析其较大的偏差属性，这些观点最终以一致性而相等，但给出了不同的观点。在第一种方法中，一个方法是从2.5级的大偏差开始，对于经验密度的关节概率和riccati过程的经验电流的概率开始，并执行适当的Euler-Lagrange优化，以计算Lyapunov指数的累积生成函数。在第二种方法中，从适当的倾斜fokker-Planck操作员的光谱分析获得了这种累积的生成函数。通过概括DOOB的H-Transform获得的相关条件过程，可以通过第一种方法阐明等效性。最后，描述了对一维安德森本地化模型的应用，以明确获取有限尺寸的Lyapunov指数的第一个累积物。
For the 2D matrix Langevin dynamics that corresponds to the continuous-time limit of the product of some $2 \times 2$ random matrices, the finite-time Lyapunov exponent can be written as an additive functional of the associated Riccati process submitted to some Langevin dynamics on the infinite periodic ring. Its large deviations properties can be thus analyzed from two points of view that are equivalent in the end by consistency but give different perspectives. In the first approach, one starts from the large deviations at level 2.5 for the joint probability of the empirical density and of the empirical current of the Riccati process and one performs the appropriate Euler-Lagrange optimization in order to compute the cumulant generating function of the Lyapunov exponent. In the second approach, this cumulant generating function is obtained from the spectral analysis of the appropriate tilted Fokker-Planck operator. The associated conditioned process obtained via the generalization of Doob's h-transform allows to clarify the equivalence with the first approach. Finally, applications to one-dimensional Anderson Localization models are described in order to obtain explicitly the first cumulants of the finite-size Lyapunov exponent.
