On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight
We study the asymptotics of recurrence coefficients for monic orthogonal polynomials πn(z) with the quartic exponential weight exp[−N(1/2z²+1/4tz⁴)], where t∈C and N∈N, N→∞. Our goal is to describe these asymptotic behaviors globally for t∈C in different regions. We also describe the ''bre...
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| Дата: | 2016 |
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| Мова: | English |
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Інститут математики НАН України
2016
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/148547 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight / M. Bertola, A. Tovbis // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. |
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irk-123456789-1485472019-02-19T01:24:20Z On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight Bertola, M. Tovbis, A. We study the asymptotics of recurrence coefficients for monic orthogonal polynomials πn(z) with the quartic exponential weight exp[−N(1/2z²+1/4tz⁴)], where t∈C and N∈N, N→∞. Our goal is to describe these asymptotic behaviors globally for t∈C in different regions. We also describe the ''breaking'' curves separating these regions, and discuss their special (critical) points. All these pieces of information combined provide the global asymptotic ''phase portrait'' of the recurrence coefficients of πn(z), which was studied numerically in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321]. The main goal of the present paper is to provide a rigorous framework for the global asymptotic portrait through the nonlinear steepest descent analysis (with the g-function mechanism) of the corresponding Riemann-Hilbert problem (RHP) and the continuation in the parameter space principle. The latter allows to extend the nonlinear steepest descent analysis from some parts of the complex t-plane to all noncritical values of t. We also provide explicit solutions for recurrence coefficients in terms of the Riemann theta functions. The leading order behaviour of the recurrence coefficients in the full scaling neighbourhoods the critical points (double and triple scaling limits) was obtained in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321] and [Asymptotics of complex orthogonal polynomials on the cross with varying quartic weight: critical point behaviour and the second Painlevé transcendents, in preparation]. 2016 Article On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight / M. Bertola, A. Tovbis // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D45; 33E17; 15B52 DOI:10.3842/SIGMA.2016.118 http://dspace.nbuv.gov.ua/handle/123456789/148547 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We study the asymptotics of recurrence coefficients for monic orthogonal polynomials πn(z) with the quartic exponential weight exp[−N(1/2z²+1/4tz⁴)], where t∈C and N∈N, N→∞. Our goal is to describe these asymptotic behaviors globally for t∈C in different regions. We also describe the ''breaking'' curves separating these regions, and discuss their special (critical) points. All these pieces of information combined provide the global asymptotic ''phase portrait'' of the recurrence coefficients of πn(z), which was studied numerically in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321]. The main goal of the present paper is to provide a rigorous framework for the global asymptotic portrait through the nonlinear steepest descent analysis (with the g-function mechanism) of the corresponding Riemann-Hilbert problem (RHP) and the continuation in the parameter space principle. The latter allows to extend the nonlinear steepest descent analysis from some parts of the complex t-plane to all noncritical values of t. We also provide explicit solutions for recurrence coefficients in terms of the Riemann theta functions. The leading order behaviour of the recurrence coefficients in the full scaling neighbourhoods the critical points (double and triple scaling limits) was obtained in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321] and [Asymptotics of complex orthogonal polynomials on the cross with varying quartic weight: critical point behaviour and the second Painlevé transcendents, in preparation]. |
| format |
Article |
| author |
Bertola, M. Tovbis, A. |
| spellingShingle |
Bertola, M. Tovbis, A. On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Bertola, M. Tovbis, A. |
| author_sort |
Bertola, M. |
| title |
On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight |
| title_short |
On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight |
| title_full |
On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight |
| title_fullStr |
On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight |
| title_full_unstemmed |
On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight |
| title_sort |
on asymptotic regimes of orthogonal polynomials with complex varying quartic exponential weight |
| publisher |
Інститут математики НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/148547 |
| citation_txt |
On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight / M. Bertola, A. Tovbis // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:29:43Z |
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