Large-N limit of horizontal Brownian motions on Lie groups

这个合作项目旨在探索N × N矩阵李群上自然水平布朗运动的大N极限。它是概率论、微分几何和群论的交叉点，特别是以一种新颖的方式将李群的随机过程研究和亚黎曼几何的研究结合起来。亚黎曼几何模型是有约束的系统，但它的建立使得系统在相空间的所有部分上运动，也就是说，约束足够灵活，空间中的任何两点都可以通过满足约束的曲线连接起来。这些类型的几何图形自然地出现在所有科学中，从机器人运动规划的受限物理系统到大脑视觉皮层第一层的建模。例如，可以通过指定车辆的中心坐标和相对于参考线的旋转角度来描述车辆在野外的位置。在这个三维参数空间中，不可能执行与车辆车轮方向垂直的运动对应的运动。然而，通过选择合适的机动，它仍然有可能到达任何目标位置。对N × N矩阵李群上布朗运动的大N极限进行了积极的研究。分析使用了自由概率的工具，其结果在随机矩阵理论中具有启示意义。在这些作品中，考虑的李群配备了一个规范的黎曼结构。我们现在计划解决一个自然的问题，即关于N x N矩阵李群上水平布朗运动的大N极限可以说些什么，在这种情况下，李群不是使用黎曼结构，而是配备了典型的亚黎曼结构。

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.