The discrete Painlevé XXXIV hierarchy arising from the gap probability distributions of Freud random matrix ensembles
Chao Min, Liwei Wang
TL;DR
This work links symmetric gap probabilities in Freud unitary ensembles to the discrete Painlevé XXXIV hierarchy by applying Chen–Ismail ladder operators to orthogonal polynomials with Freud weights on the complement of a symmetric interval. For $m=1,2,3$, the recurrence coefficients $\beta_n(a)$ satisfy higher-order nonlinear difference equations that, after a simple change of variables, realize the discrete Painlevé XXXIV hierarchy, and are accompanied by corresponding differential-difference equations for $\beta_n(a)$ and expressions for the log-derivative $\sigma_n(a)=a\frac{d}{da}\ln\mathbb{P}(n,a)$. The paper also derives explicit relations among $\beta_n$, the auxiliary quantities $R_n(a)$ and $r_n(a)$, and the leading coefficients $p(n,a)$ of the monic orthogonal polynomials, clarifying how gap probabilities relate to the nonlinear integrable structure. A key contribution is demonstrating the first appearance of the discrete Painlevé XXXIV hierarchy in Random Matrix Theory and outlining a path to generalizing to higher $m$ with anticipated $(2m)$-th order equations. The results pave the way for asymptotic analyses and further connections between orthogonal polynomials, Hankel determinants, and discrete Painlevé systems in spectral gap problems.
Abstract
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.