High-dimensional metric-measure limit of Stiefel and Grassmann manifolds

TL;DR

This paper investigates the high-dimensional limits of Stiefel and Grassmann manifolds as metric measure spaces, revealing they converge to Gaussian spaces or their quotients, and provides asymptotic estimates of their observable diameters.

Contribution

It characterizes the metric measure limits of Stiefel and Grassmann manifolds in high dimensions, connecting them to Gaussian spaces and their quotients, which is a novel insight.

Findings

Limits are Gaussian spaces or quotients in Gromov's topology.

Provides asymptotic estimates of observable diameters.

Highlights drastic differences from finite-dimensional manifolds.

Abstract

We study the high-dimensional limit of (projective) Stiefel and Grassmann manifolds as metric measure spaces in Gromov's topology. The limits are either the infinite-dimensional Gaussian space or its quotient by an mm-isomorphic group action, which are drastically different from the manifolds. As a corollary, we obtain some asymptotic estimates of the observable diameter of (projective) Stiefel and Grassmann manifolds.