10 research outputs found
Laplace approximation of Lauricella functions F A and F D
The Lauricella functions, which are generalizations of the Gauss hypergeometric function 2 F 1, arise naturally in many areas of mathematics and statistics. So far as we are aware, there is little or nothing in the literature on how to calculate numerical approximations for these functions outside those cases in which a simple one-dimensional integral representation or a one-dimensional series representation is available. In this paper we present first-order and second-order Laplace approximations to the Lauricella functions F(n)A and F(n)D. Our extensive numerical results show that these approximations achieve surprisingly good accuracy in a wide variety of examples, including cases well outside the asymptotic framework within which the approximations were derived. Moreover, it turns out that the second-order Laplace approximations are usually more accurate than their first-order versions. The numerical results are complemented by theoretical investigations which suggest that the approximations have good relative error properties outside the asymptotic regimes within which they were derived, including in certain cases where the dimension n goes to infinity
Matrix Models for the Black Hole Information Paradox
We study various matrix models with a charge-charge interaction as toy models
of the gauge dual of the AdS black hole. These models show a continuous
spectrum and power-law decay of correlators at late time and infinite N,
implying information loss in this limit. At finite N, the spectrum is discrete
and correlators have recurrences, so there is no information loss. We study
these models by a variety of techniques, such as Feynman graph expansion, loop
equations, and sum over Young tableaux, and we obtain explicitly the leading
1/N^2 corrections for the spectrum and correlators. These techniques are
suggestive of possible dual bulk descriptions. At fixed order in 1/N^2 the
spectrum remains continuous and no recurrence occurs, so information loss
persists. However, the interchange of the long-time and large-N limits is
subtle and requires further study.Comment: 35 pages, 11 eps figures; v.2 minor typos fixe