The discrete Painlevé XXXIV hierarchy arising from the gap probability distributions of Freud unitary ensembles

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Summary

The discrete Painlevé XXXIV hierarchy is derived from the gap probability distributions of Freud unitary ensembles. The recurrence coefficients of orthogonal polynomials with respect to a modified Freud weight are studied, and the relationship between the recurrence coefficients and the discrete Painlevé XXXIV hierarchy is established for m = 1, 2, and 3 cases.

Highlights

• The discrete Painlevé XXXIV hierarchy is linked to the gap probability distributions of Freud unitary ensembles.
• The recurrence coefficients of orthogonal polynomials with respect to a modified Freud weight are studied.
• The relationship between the recurrence coefficients and the discrete Painlevé XXXIV hierarchy is established for m = 1, 2, and 3 cases.
• The recurrence coefficients satisfy a second-order nonlinear difference equation for m = 1.
• The recurrence coefficients satisfy a fourth-order nonlinear difference equation for m = 2.
• The recurrence coefficients satisfy a sixth-order nonlinear difference equation for m = 3.
• The results can be used to study the large n asymptotic behavior of the recurrence coefficients and the gap probability distributions.

Key Insights

• The study of the gap probability distributions of Freud unitary ensembles leads to the derivation of the discrete Painlevé XXXIV hierarchy, which is a significant result in the field of random matrix theory and orthogonal polynomials.
• The recurrence coefficients of orthogonal polynomials with respect to a modified Freud weight are crucial in establishing the relationship between the recurrence coefficients and the discrete Painlevé XXXIV hierarchy.
• The results of this study can be used to investigate the large n asymptotic behavior of the recurrence coefficients and the gap probability distributions, which is an important area of research in random matrix theory.
• The discrete Painlevé XXXIV hierarchy is a powerful tool for studying the properties of orthogonal polynomials and random matrix ensembles, and this study provides a new perspective on this topic.
• The relationship between the recurrence coefficients and the discrete Painlevé XXXIV hierarchy is established for m = 1, 2, and 3 cases, which provides a solid foundation for further research in this area.
• The study of the gap probability distributions of Freud unitary ensembles has important implications for various fields, including physics, mathematics, and computer science.
• The results of this study can be used to develop new methods and techniques for studying the properties of orthogonal polynomials and random matrix ensembles.

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Citation

Min, C., & Wang, L. (2024). The discrete Painlevé XXXIV hierarchy arising from the gap probability distributions of Freud unitary ensembles (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.18782