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We show that the one-parameter family of special solutions of P \(_{\text {II}}\) , the second Painlevé equation, constructed from the Airy functions, as well as associated solutions of P \(_{\text {XXXIV}}\) and S \(_{\text {II}}\) , can be expressed via the recurrence coefficients of orthogonal polynomials that appear in the analysis of the Hermitian random matrix ensemble with a cubic potential. Exploiting this connection we show that solutions of P \(_{\text {II}}\) that depend only on the first Airy function \( {\text {Ai}}\) (but not on \( {\text {Bi}}\) ) possess a scaling limit in the pole free region, which includes a disk around the origin whose radius grows with the parameter. We then use the scaling limit to show that these solutions are monotone in the parameter on the negative real axis.

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On Airy Solutions of P \(_{\text {II}}\) and Complex Cubic Ensemble of Random Matrices, I

  • 会议论文

  • 发表日期：

    2025年

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  • Ahmad Barhoumi,
  • Pavel Bleher,
  • Alfredo Deaño,
  • Maxim Yattselev

摘要

We show that the one-parameter family of special solutions of P \(_{\text {II}}\) , the second Painlevé equation, constructed from the Airy functions, as well as associated solutions of P \(_{\text {XXXIV}}\) and S \(_{\text {II}}\) , can be expressed via the recurrence coefficients of orthogonal polynomials that appear in the analysis of the Hermitian random matrix ensemble with a cubic potential. Exploiting this connection we show that solutions of P \(_{\text {II}}\) that depend only on the first Airy function \( {\text {Ai}}\) (but not on \( {\text {Bi}}\) ) possess a scaling limit in the pole free region, which includes a disk around the origin whose radius grows with the parameter. We then use the scaling limit to show that these solutions are monotone in the parameter on the negative real axis.