The Generalised Dyson Circular Unitary Ensemble: Asymptotic Distribution of the Eigenvalues at the Origin of the Spectrum
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- 作者：Philippe Rambour (1) philippe.rambour@math.u-psud.fr
Abdellatif Seghier (1) abdelatif.seghier@math.u-psud.fr - 关键词：Random unitary matrices – ; orthogonal polynomials – ; Fisher ; Hartwig conjecture
- 刊名：Integral Equations and Operator Theory
- 出版时间：April 2011
- 出版年：2011
- 期刊代码：64_0378-620X
- 类别：mat
- 卷：69
- 期：4
- 页码：535-555
- 数据来源：sp
摘要
The first part of this paper is devoted to the study of FN{\Phi_N} the orthogonal polynomials on the circle, with respect to a weight of type f = (1 − cos θ) α c where c is a sufficiently smooth function and ${\alpha > -\frac{1}{2}}${\alpha > -\frac{1}{2}}. We obtain an asymptotic expansion of the coefficients F*(p)N(1){\Phi^{*(p)}_{N}(1)} for all integer p where F*N{\Phi^*_N} is defined by F*N (z) = zN [`(F)]N(\frac1z) (z 鹿 0){\Phi^*_N (z) = z^N \bar \Phi_N(\frac{1}{z})\ (z \not=0)}. These results allow us to obtain an asymptotic expansion of the associated Christofel–Darboux kernel, and to compute the distribution of the eigenvalues of a family of random unitary matrices. The proof of the results related to the orthogonal polynomials are essentially based on the inversion of the Toeplitz matrix associated to the symbol f.