Asymptotics for products of characteristic polynomials in classical β\beta-Ensembles

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 scaling limits of the expectation value of products of characteristic polynomials as $N\to\infty$. In the bulk of the spectrum of each $\beta$-ensemble, the same scaling limit is found to be $e^{p_{1}}{}_1F_{1}$ whose exact expansion in terms of Jack polynomials is well known. The scaling limit at the soft edge of the spectrum for 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 even, scaling limits of the $k$-point correlation functions for the three ensembles are 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 integrals of Selberg type.Comment: [v3] 35 pages; this is a revised and enlarged version of the article with new references, simplified demonstations, and improved presentation. To be published in Constructive Approximation 37 (2013

Multifrequency variability of the blazar AO 0235+164 the WEBT campaign in 2004-2005 and long-term SED analysis

A huge multiwavelength campaign targeting the blazar AO 0235+164 was organized by the Whole Earth Blazar Telescope (WEBT) in 2003-2005 to study the variability properties of the source. Monitoring observations were carried out at cm and mm wavelengths, and in the near-IR and optical bands, while three pointings by the XMM-Newton satellite provided information on the X-ray and UV emission. We present the data acquired during the second observing season, 2004-2005, by 27 radio-to-optical telescopes. They reveal an increased near-IR and optical activity with respect to the previous season. Increased variability is also found at the higher radio frequencies, down to 15 GHz, but not at the lower ones. The radio (and optical) outburst predicted to peak around February-March 2004 on the basis of the previously observed 5-6 yr quasi-periodicity did not occur. The analysis of the optical light curves reveals now a longer characteristic time scale of 8 yr, which is also present in the radio data. The spectral energy distributions corresponding to the XMM-Newton observations performed during the WEBT campaign are compared with those pertaining to previous pointings of X-ray satellites. Bright, soft X-ray spectra can be described in terms of an extra component, which appears also when the source is faint through a hard UV spectrum and a curvature of the X-ray spectrum. Finally, there might be a correlation between the X-ray and optical bright states with a long time delay of about 5 yr, which would require a geometrical interpretation