Large-N limit of horizontal Brownian motions on Lie groups

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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.