Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 013, 18 pages      arXiv:1409.4213      https://doi.org/10.3842/SIGMA.2015.013

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

Margit Rösler a and Michael Voit b
a) Institut für Mathematik, Universität Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany
b) Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany

Received October 14, 2014, in final form February 03, 2015; Published online February 10, 2015

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.

Key words: Mehler-Heine formula; Heckman-Opdam polynomials; Grassmann manifolds; spherical functions; central limit theorem; asymptotic representation theory.

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