Some asymptotic properties of the spectrum of the Jacobi ensemble

Details Some asymptotic properties of the spectrum of the Jacobi ensemble Some asymptotic properties of the spectrum of the Jacobi ensemble

2009-04-27

arxiv.org/abs/0904.4091v1

Mathematics

Probability

20 pages, 2 figures

Scientific paper

For the random eigenvalues with density corresponding to the Jacobi ensemble $$c \cdot \prod_{i < j} | \lambda_i - \lambda_j |^\beta \prod^n_{i=1} (2 - \lambda_i)^a (2 + \lambda_i)^b I_{(-2,2)} (\lambda_i) $$ $(a, b > -1, \beta > 0) $ a strong uniform approximation by the roots of the Jacobi polynomials is derived if the parameters $a, b,$ $\beta$ depend on $n$ and $n \to \infty$. Roughly speaking, the eigenvalues can be uniformly approximated by roots of Jacobi polynomials with parameters $((2a+2)/\beta -1, (2b+2)/\beta-1)$, where the error is of order $\{\log n/(a+b) \}^{1/4}$. These results are used to investigate the asymptotic properties of the corresponding spectral distribution if $n \to \infty$ and the parameters $a, b$ and $\beta$ vary with $n$. We also discuss further applications in the context of multivariate random $F$-matrices.

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