The Hankel Determinants from a Singularly Perturbed Jacobi Weight

The Hankel Determinants from a Singularly Perturbed Jacobi Weight

We study the Hankel determinant generated by a singularly perturbed Jacobi weight <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mrow><mo>(</mo><mi>x</mi&...

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Autores principales: Pengju Han, Yang Chen
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
random matrix theory
Hankel determinant
singularly perturbed Jacobi weight
ladder operators
Painlevé V
Mathematics
QA1-939
Acceso en línea:https://doaj.org/article/d6b7b74391b445a7b9ac56a998206092
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institution DOAJ
collection DOAJ
language EN
topic random matrix theory
Hankel determinant
singularly perturbed Jacobi weight
ladder operators
Painlevé V
Mathematics
QA1-939
spellingShingle random matrix theory
Hankel determinant
singularly perturbed Jacobi weight
ladder operators
Painlevé V
Mathematics
QA1-939
Pengju Han
Yang Chen
The Hankel Determinants from a Singularly Perturbed Jacobi Weight
description We study the Hankel determinant generated by a singularly perturbed Jacobi weight <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mi>α</mi></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></mrow><mi>β</mi></msup><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><mfrac><mi>s</mi><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac></mrow></msup><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>x</mi><mo>∈</mo><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>α</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>β</mi><mo>></mo><mn>0</mn><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>s</mi><mo>≥</mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, it is reduced to the classical Jacobi weight. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, the factor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><mfrac><mi>s</mi><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac></mrow></msup></semantics></math></inline-formula> induces an infinitely strong zero at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. For the finite <i>n</i> case, we obtain four auxiliary quantities <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>r</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>R</mi><mo>˜</mo></mover><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>r</mi><mo>˜</mo></mover><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> by using the ladder operator approach. We show that the recurrence coefficients are expressed in terms of the four auxiliary quantities with the aid of the compatibility conditions. Furthermore, we derive a shifted Jimbo–Miwa–Okamoto <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-function of a particular Painlevé V for the logarithmic derivative of the Hankel determinant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. By variable substitution and some complicated calculations, we show that the quantity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> satisfies the four Painlevé equations. For the large <i>n</i> case, we show that, under a double scaling, where <i>n</i> tends to <i>∞</i> and <i>s</i> tends to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>0</mn><mo>+</mo></msup></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>:</mo><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup><mi>s</mi></mrow></semantics></math></inline-formula> is finite, the scaled Hankel determinant can be expressed by a particular <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mrow><mi>I</mi><mi>I</mi><msup><mi>I</mi><mo>′</mo></msup></mrow></msub></semantics></math></inline-formula>.
format article
author Pengju Han
Yang Chen
author_facet Pengju Han
Yang Chen
author_sort Pengju Han
title The Hankel Determinants from a Singularly Perturbed Jacobi Weight
title_short The Hankel Determinants from a Singularly Perturbed Jacobi Weight
title_full The Hankel Determinants from a Singularly Perturbed Jacobi Weight
title_fullStr The Hankel Determinants from a Singularly Perturbed Jacobi Weight
title_full_unstemmed The Hankel Determinants from a Singularly Perturbed Jacobi Weight
title_sort hankel determinants from a singularly perturbed jacobi weight
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/d6b7b74391b445a7b9ac56a998206092
work_keys_str_mv AT pengjuhan thehankeldeterminantsfromasingularlyperturbedjacobiweight
AT yangchen thehankeldeterminantsfromasingularlyperturbedjacobiweight
AT pengjuhan hankeldeterminantsfromasingularlyperturbedjacobiweight
AT yangchen hankeldeterminantsfromasingularlyperturbedjacobiweight
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spelling oai:doaj.org-article:d6b7b74391b445a7b9ac56a9982060922021-11-25T18:17:48ZThe Hankel Determinants from a Singularly Perturbed Jacobi Weight10.3390/math92229782227-7390https://doaj.org/article/d6b7b74391b445a7b9ac56a9982060922021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2978https://doaj.org/toc/2227-7390We study the Hankel determinant generated by a singularly perturbed Jacobi weight <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mi>α</mi></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></mrow><mi>β</mi></msup><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><mfrac><mi>s</mi><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac></mrow></msup><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>x</mi><mo>∈</mo><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>α</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>β</mi><mo>></mo><mn>0</mn><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>s</mi><mo>≥</mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, it is reduced to the classical Jacobi weight. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, the factor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><mfrac><mi>s</mi><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac></mrow></msup></semantics></math></inline-formula> induces an infinitely strong zero at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. For the finite <i>n</i> case, we obtain four auxiliary quantities <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>r</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>R</mi><mo>˜</mo></mover><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>r</mi><mo>˜</mo></mover><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> by using the ladder operator approach. We show that the recurrence coefficients are expressed in terms of the four auxiliary quantities with the aid of the compatibility conditions. Furthermore, we derive a shifted Jimbo–Miwa–Okamoto <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-function of a particular Painlevé V for the logarithmic derivative of the Hankel determinant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. By variable substitution and some complicated calculations, we show that the quantity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> satisfies the four Painlevé equations. For the large <i>n</i> case, we show that, under a double scaling, where <i>n</i> tends to <i>∞</i> and <i>s</i> tends to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>0</mn><mo>+</mo></msup></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>:</mo><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup><mi>s</mi></mrow></semantics></math></inline-formula> is finite, the scaled Hankel determinant can be expressed by a particular <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mrow><mi>I</mi><mi>I</mi><msup><mi>I</mi><mo>′</mo></msup></mrow></msub></semantics></math></inline-formula>.Pengju HanYang ChenMDPI AGarticlerandom matrix theoryHankel determinantsingularly perturbed Jacobi weightladder operatorsPainlevé VMathematicsQA1-939ENMathematics, Vol 9, Iss 2978, p 2978 (2021)

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