The discrete Painlevé XXXIV hierarchy arising from the gap probability distributions of Freud random matrix ensembles
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| Format: | Preprint |
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2024
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| _version_ | 1866916567837573120 |
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| author | Min, Chao Wang, Liwei |
| author_facet | Min, Chao Wang, Liwei |
| contents | We consider the symmetric gap probability distributions of certain Freud unitary ensembles. This problem is related to the Hankel determinants generated by the Freud weights supported on the complement of a symmetric interval. By using Chen and Ismail's ladder operator approach, we obtain the difference equations satisfied by the recurrence coefficients for the orthogonal polynomials with the discontinuous Freud weights. We find that these equations, with a minor change of variables, are the discrete Painlevé XXXIV hierarchy proposed by Cresswell and Joshi [{\em J. Phys. A: Math. Gen.} {\bf 32} ({1999}) {655--669}]. This is the first time that the discrete Painlevé XXXIV hierarchy appears in the study of Random Matrix Theory. We also derive the differential-difference equations for the recurrence coefficients and show the relationship between the logarithmic derivative of the gap probabilities, the nontrivial leading coefficients of the monic orthogonal polynomials and the recurrence coefficients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_18782 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The discrete Painlevé XXXIV hierarchy arising from the gap probability distributions of Freud random matrix ensembles Min, Chao Wang, Liwei Exactly Solvable and Integrable Systems Mathematical Physics 60B20, 42C05 We consider the symmetric gap probability distributions of certain Freud unitary ensembles. This problem is related to the Hankel determinants generated by the Freud weights supported on the complement of a symmetric interval. By using Chen and Ismail's ladder operator approach, we obtain the difference equations satisfied by the recurrence coefficients for the orthogonal polynomials with the discontinuous Freud weights. We find that these equations, with a minor change of variables, are the discrete Painlevé XXXIV hierarchy proposed by Cresswell and Joshi [{\em J. Phys. A: Math. Gen.} {\bf 32} ({1999}) {655--669}]. This is the first time that the discrete Painlevé XXXIV hierarchy appears in the study of Random Matrix Theory. We also derive the differential-difference equations for the recurrence coefficients and show the relationship between the logarithmic derivative of the gap probabilities, the nontrivial leading coefficients of the monic orthogonal polynomials and the recurrence coefficients. |
| title | The discrete Painlevé XXXIV hierarchy arising from the gap probability distributions of Freud random matrix ensembles |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics 60B20, 42C05 |
| url | https://arxiv.org/abs/2412.18782 |