English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We obtain the strong asymptotics of polynomials $p_n(\lambda)$,$\lambda\in\mathbb{C}$, orthogonal with respect to measures in the complexplane of the form $$e^{-N(|\lambda|^{2s}-t\lambda^s-\overline{t\lambda}^s)}dA(\lambda), $$ where$s$ is a positive integer, $t$ is a complex parameter and $dA$ stands for thearea measure in the plane. Such problem has its origin from normal matrixmodels. We study the asymptotic behaviour of $p_n(\lambda)$ in the limit$n,N\to\infty$ in such a way that $n/N\to T$ constant. Such asymptoticbehaviour has two distinguished regimes according to the topology of thelimiting support of the eigenvalue distribution of the normal matrix model. If$0<|t|^2<T/s$, the eigenvalue distribution support is a simply connectedcompact set of the complex plane, while for $|t|^2>T/s$ the eigenvaluedistribution support consists of $s$ connected components. Correspondingly thesupport of the limiting zero distribution of the orthogonal polynomialsconsists of a closed contour contained in each connected component. Ourasymptotic analysis is obtained by reducing the planar orthogonality conditionsof the polynomials to an equivalent system of contour integral orthogonalityconditions. The strong asymptotics for the orthogonal polynomials is obtainedfrom the corresponding Riemann--Hilbert problem by the Deift--Zhou nonlinearsteepest descent method.

Orthogonal polynomials for a class of measures with discrete rotational symmetries in the complex plane