Large-N limit of horizontal Brownian motions on Lie groups

这个合作项目旨在探索自然水平布朗运动在N×N矩阵李群上的大-N极限。它位于概率论、微分几何和群论的交叉点上，特别是以一种新颖的方式结合了李群上的随机过程的研究和次黎曼几何的研究，建立了具有约束的次黎曼几何模型系统，即约束足够灵活，使得空间中的任何两个点都可以由满足约束的曲线连接。这些类型的几何图形自然出现在所有科学中，从受限的物理系统到机器人的运动规划，再到对大脑视觉皮质第一层进行建模。例如，车辆在场地中的位置可以通过指定其中心坐标和相对于基准线的旋转角度来描述。在该三维参数空间中，不可能执行对应于垂直于车辆车轮方向的运动的运动。然而，通过选择适当的机动，仍然有可能到达任何目标位置。关于N×N矩阵李群上的布朗运动的大-N极限已被活跃地研究。该分析使用了自由概率中的工具，其结果在随机矩阵理论中具有一定的意义。在这些工作中，所考虑的李群配备了典型的黎曼结构。我们现在计划解决关于N×N矩阵的李群上的水平布朗运动的大-N极限可以说是什么的自然问题，其中，李群不使用黎曼结构，而是配备了典型的次黎曼结构。

This collaborative project aims to explore large-N limits of natural horizontal Brownian motions on Lie groups of N x N matrices. It lies at the intersection of probability theory, differential geometry and group theory, and particularly combines the study of stochastic processes on Lie groups and the study of sub-Riemannian geometries in a novel way.Sub-Riemannian geometries model systems with constraints but set up such that the system moves over all parts of the phase space, that is, the constraints are flexible enough that any two points in the space can be connected by a curve satisfying the constraints. These type of geometries naturally appear in all sciences, ranging from constrained physical systems over motion planning in robotics to modelling the first layer of the visual cortex of the brain. For instance, the position of a vehicle in a field can be described by specifying the coordinates of its centre and the angle of rotation with respect to a reference line. In this three-dimensional parameter space, it is not possible to perform motions which correspond to a movement perpendicular to the direction of the wheels of the vehicle. However, by choosing suitable maneuvers it is still possible to reach any target position.Large-N limits of Brownian motions on Lie groups of N x N matrices have been actively studied. The analysis employs tools from free probability and the results have implications in random matrix theory. In these works, the Lie groups in considerations are equipped with a canonical Riemannian structure.We plan to now tackle the natural question of what can be said about the large-N limits of horizontal Brownian motions on Lie groups of N x N matrices where, instead of using a Riemannian structure, the Lie groups are equipped with canonical sub-Riemannian structures.