A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian

TL;DR

This paper establishes a central limit theorem for random walks on the dual of compact Grassmannians, connecting harmonic analysis, special functions, and random matrix theory, with explicit error estimates and broader parameter applicability.

Contribution

It introduces a sharp Mehler-Heine formula for Heckman-Opdam polynomials and applies it to derive a new CLT for tensor power decompositions, extending to a larger parameter range.

Findings

Derived a precise Mehler-Heine approximation for Heckman-Opdam polynomials.

Proved a CLT for random walks on the dual of Grassmannians.

Connected the limit distribution to Laguerre ensembles in random matrix theory.

Abstract

We consider compact Grassmann manifolds $G/K$ over the real, complex or quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of type $BC$. From an explicit integral representation of these polynomials we deduce a sharp Mehler-Heine formula, that is an approximation of the Heckman-Opdam polynomials in terms of Bessel functions, with a precise estimate on the error term. This result is used to derive a central limit theorem for random walks on the semi-lattice parametrizing the dual of $G/K$, which are constructed by successive decompositions of tensor powers of spherical representations of $G$. The limit is the distribution of a Laguerre ensemble in random matrix theory. Most results of this paper are established for a larger continuous set of multiplicity parameters beyond the group cases.