Physics – Mathematical Physics
Scientific paper
2011-12-05
Physics
Mathematical Physics
25 pages
Scientific paper
We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral) and Jacobi $\beta$-ensembles of $N\times N$ random matrices. More specifically, we calculate the scaling limit of the expectation value of the product of $n$ characteristic polynomials of these random matrices as $N\to\infty$. In the bulk of the spectrum of each $\beta$-ensemble, the same limiting expectation is found: a multivariate hypergeometric function ${}_1F^{(\beta/2)}_{1}$ whose exact expansion in terms of Jack polynomials is well-known. At the hard edge of spectrum of the Laguerre and Jacobi $\beta$-ensembles, the limit is the ${}_0F^{(\beta/2)}_{1}$ multivariate hypergeometric function. Finally, the limiting expectation at the soft edge of the spectrum of the Hermite and Laguerre $\beta$-ensembles is shown to be a multivariate Airy function, which is defined as a generalized Kontsevich integral. As corollaries, when $\beta$ is an even integer the limit of point-correlation functions for the three ensembles is obtained. The asymptotics of the multivariate Airy function for large and small arguments is also given. All the asymptotic results rely on a generalization of Watson's lemma and the steepest descent method for higher dimensional integrals of Selberg type.
Desrosiers Patrick
Liu Dang-Zheng
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