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

作者：Roesler Margit; Voit Michael

来源：Symmetry Integrability and Geometry-Methods and Applications, 2015, 11: 013.

DOI：10.3842/SIGMA.2015.013

• Mehler-Heine formula
• Heckman-Opdam polynomials
• Grassmann manifolds
• spherical functions
• central limit theorem
• asymptotic representation theory

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.

• 出版日期2015