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

Symmetry Integrability and Geometry-Methods and Applications

article

Margit Rösler¹  Michael Voit² 

[1] Institut für Mathematik;Fakultät für Mathematik, Technische Universität Dortmund

关键词: Mehler–Heine formula;    Heckman–Opdam polynomials;    Grassmann manifolds;    spherical functions;    central limit theorem;    asymptotic representation theory;   

DOI  :  10.3842/SIGMA.2015.013

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown