Riemann-Hilbert hierarchies for hard edge planar orthogonal polynomials
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Published:2024-04
Issue:2
Volume:146
Page:371-403
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ISSN:1080-6377
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Container-title:American Journal of Mathematics
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language:en
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Short-container-title:ajm
Author:
Hedenmalm Haakan,Wennman Aron
Abstract
abstract: We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $\mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly positive, and for any given precision $\varkappa$, the expansion holds with an $\mathrm{O}(N^{-\varkappa-1})$ error in $N$-dependent neighborhoods of the exterior region as the degree $N$ tends to infinity. The main ingredient is the derivation and analysis of Riemann-Hilbert hierarchies---sequences of scalar Riemann-Hilbert problems---which allows us to express all higher order correction terms in closed form. Indeed, the expansion may be understood as a Neumann series involving an explicit operator. The expansion theorem leads to a semiclassical asymptotic expansion of the corresponding hard edge probability wave function in terms of distributions supported on $\partial\mathscr{D}$.
Cited by
1 articles.
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