Ewens measures on compact groups and hypergeometric kernels

Details Ewens measures on compact groups and hypergeometric kernels Ewens measures on compact groups and hypergeometric kernels

2007-12-05

arxiv.org/abs/0712.0848v2

Mathematics

Probability

New version of the previous paper "Hua-Pickrell measures on general compact groups". The article has been completely re-writte

Scientific paper

On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure, these factors become independent random variables with explicit distributions. Beyond the known results on the orthogonal and unitary groups (O(n) and U(n)), we treat the symplectic case. In U(n), this induces a family of probability changes analogous to the biassing in the Ewens sampling formula known for the symmetric group. Then we study the spectral properties of these measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The associated orthogonal polynomials give rise, as $n$ tends to infinity to a limit kernel at the singularity.

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