Local limits for orthogonal polynomials for varying measures via universality
In the setting of varying measures, the authors establish local asymptotics from universality limits. More precisely, they consider orthogonal polynomials \(\left\{ p_n \left(e^{-2nQ_n}, x \right) \right\}\) and use universality limits to prove ``local limits'' \[ \lim_{n \to +\infty} \frac{ p_n \left(e^{-2nQ_n}, y_{jn} + \frac{z}{\tilde{K}_{n} \left(y_{jn}, y_{jn} \right)} \right)}{ p_n \left(e^{-2nQ_n}, y_{jn} \right)} \, e^{{}^{-\frac{nQ_{n}'(y_{jn})}{\tilde{K}_{n} \left(y_{jn}, y_{jn} \right)}z}} = \cos(\pi z), \] where \(y_{jn}\) is a local maximum point of \(|p_n| e^{-n Q_n}\) in the ``bulk'' of the support, \(\tilde{K}_{n} \left(y_{jn}, y_{jn} \right)\) is the normalized reproducing kernel, and the limit holds uniformly for \(z\) in compact subsets of the plane. They also consider local limits at the ``soft edge'' of the spectrum, which involve the Airy function. Unlike from earlier works (see e.g. [\textit{D. S. Lubinsky}, Proc. Am. Math. Soc. 147, No. 9, 3877--3886 (2019; Zbl 1422.42037)]), in this paper the authors consider extra factors in the limits.
scientific article; zbMATH DE number 7187599
scientific article