2024/12/25 by Chao Min, Liwei Wang, Min, Chao +1
Mathematics · Physics and Astronomy · #42C05 #60B20 #Advanced Statistical Methods and Models #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Fuzzy Systems and Optimization #Mathematical Physics (math-ph) #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2412.18782
openalex publication_date 2024/12/25 · openalex created_date 2024/12/31 · openalex updated_date 2026/07/28
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.