Pointwise Asymptotics for Orthonormal Polynomials at the Endpoints of the Interval via Universality

We show that universality limits and other bounds imply pointwise asymptotics for orthonormal polynomials at the endpoints of the interval of orthonormality. As a consequence, we show that if $\mu $ is a regular measure supported on $\left [ -1,1\right ] $, and in a neighborhood of 1, $ \mu $ is absolutely continuous, while for some $\alpha>-1$, $\mu ^{\prime }\left ( t\right ) =h\left ( t\right ) \left ( 1-t\right )^{\alpha }$, where $ h\left ( t\right ) \rightarrow 1$ as $t\rightarrow 1-$, then the corresponding orthonormal polynomials $\left \{ p_{n}\right \} $ satisfy the asymptotic $$ \lim_{n\rightarrow \infty }\frac{p_{n}\left( 1-\frac{z^{2}}{2n^{2}}\right) }{ p_{n}\left( 1\right) }=\frac{J_{\alpha }^{\ast }\left( z\right) }{J_{\alpha }^{\ast }\left( 0\right) } $$uniformly in compact subsets of the plane. Here $J_{\alpha }^{\ast }\left ( z\right ) =J_{\alpha }\left ( z\right ) /z^{\alpha }$ is the normalized Bessel function of order $\alpha $. These are by far the most general conditions for such endpoint asymptotics.

DISCRETE CIRCULAR BETA ENSEMBLES

Let μ be a measure with support on the unit circle and n ≥ 1, β > 0. The associated circular β ensemble involves a probability distribution of the form [Formula: see text] where C is a normalization constant, and [Formula: see text] We explicitly evaluate the m-point correlation functions when μ is replaced by a discrete measure on the unit circle, generated by paraorthogonal orthogonal polynomials associated with μ, and use this to investigate universality limits for sequences of such measures. We also consider ratios of products of random characteristic polynomials.