A New Approach to Universality Limits Involving Orthogonal Polynomials

By Doron S. Lubinsky. 2007

182 citations recorded by the source corpus.

We show how localization and smoothing techniques can be used to establish universality in the bulk of the spectrum for a fixed positive measure /x on [? 1,1]. Assume that ji is a regular measure, and is absolutely continuous in an open interval containing some point x. Assume moreover, that /z' is positive and continuous at x. Then universality for /x holds at x. If the hypothesis holds for x in a compact subset of (?1,1), universality holds uniformly for such x. Indeed, this follows from universality for the classical Legendre weight. We also establish universality in an Lp sense under weaker assumptions on fi.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9cc3ac959795b330d50d67c275bfc3b11ef3d0d9

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In Applied Mathematics

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Citations in the corpus

A Dynamical Approach to Random Matrix Theory (2017, methodology): “An alternative method was presented in [98,101] using more direct methods from orthogonal polynomials.”
Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials (2013, background): “Proof This is essentially the same as in [8], but we include the details because of the different setting.”
Universality limits in the bulk for varying measures (2008, unknown): “Then (5.6) ` 1X j=k+1 n (tjn) e Btjn Z t`n tkn e Btd n (t) X̀ j=k n (tjn) e Btjn :”
Universality and fine zero spacing on general sets (2009, methodology): “Among the highlights of this new approach is universality under the sole continuity of the weight [ 12 ] (previously analyticity was required!), fine spacing of zeros of orthogonal polynomials by Levin and Lubinsky in [10], and universality for exponential weights in [11].”
On a problem by Steklov (2014, background): “mial entropies – another important quantity to measure the size of the polynomial. Remark. The size, asymptotics, and universality of the Christoﬀel-Darboux kernel were extensively studied, see, e.g. [17, 16, 18, 28, 29]. We, however, will focus on φn itself. Remark. Since Sδ is invariant under the ”
Universality of local spectral statistics of random matrices (2011, background): “The extension of this approach to a general potential is a demanding task; important progress was made since the late 1990s by Fokas, Its, and Kitaev [42], Bleher and Its [8], Deift et al. [14, 17, 18], Pastur and Shcherbina [55, 56], and more recently by Lubinsky [50].”
Universality of Wigner random matrices: a survey of recent results (2010, methodology): “An alternative method was presented in [68, 67] using more direct methods from orthogonal polynomials.”