Generalized Christoffel-Darboux formula for classical skew-orthogonal polynomials
Abstract
We show that skew-orthogonal functions, defined with respect to Jacobi weight $w_{a,b}(x)={(1-x)}^a{(1+x)}^b$, $a$, $b>-1$, including the limiting cases of Laguerre ($w_{a}(x)=x^{a}e^{-x}$, $a > -1$) and Gaussian weight ($w(x)=e^{-x^2}$), satisfy three-term recursion relation in the quaternion space. From this, we derive generalized Christoffel-Darboux (GCD) formulæ for kernel functions arising in the study of the corresponding orthogonal and symplectic ensembles of random $2N\times 2N$ matrices. Using the GCD formulæwe calculate the level-densities and prove that in the bulk of the spectrum, under appropriate scaling, the eigenvalue correlations are universal. We also provide evidence to show that there exists a mapping between skew-orthogonal functions arising in the study of orthogonal and symplectic ensembles of random matrices.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2007
- DOI:
- 10.48550/arXiv.0711.4432
- arXiv:
- arXiv:0711.4432
- Bibcode:
- 2007arXiv0711.4432S
- Keywords:
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- Mathematical Physics
- E-Print:
- 29 pages