Gaussian unitary ensembles with jump discontinuities, PDEs, and the coupled Painlevé IV system

Abstract

We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t 1"> <mml:semantics> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">t_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t Subscript m"> <mml:semantics> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">t_m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary conditions, we show that the logarithmic derivative of the Hankel determinant satisfies a second-order partial differential equation which is reduced to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-form of a Painlevé IV equation when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals 1"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, under the assumption that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t Subscript k Baseline minus t 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">t_k-t_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is fixed for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 2"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, by considering the Riemann-Hilbert problem for the orthogonal polynomials, we construct direct relationships between the auxiliary quantities introduced in the ladder operators and solutions of a coupled Painlevé IV system.

Locations

• Contemporary mathematics - American Mathematical Society - View
• arXiv (Cornell University) - View - PDF