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arXiv · 1409.4213

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

Margit Rösler·Michael Voit
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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.

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Margit Rösler, Michael Voit. 2015-02-10. A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian. https://doi.org/10.3842/sigma.2015.013

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