Painlevé V, Painlevé XXXIV and the Degenerate Laguerre Unitary Ensemble

Abstract

In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble. This problem originates from the largest or smallest eigenvalue distribution of the degenerate Laguerre unitary ensemble. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight. By applying the ladder operators to our problem, we obtain two auxiliary quantities $R_n(t)$ and $r_n(t)$ and show that they satisfy the coupled Riccati equations, from which we find that $R_n(t)$ satisfies the Painlevé V equation. Furthermore, we prove that ${\sigma }_n(t)$, a quantity related to the logarithmic derivative of the Hankel determinant, satisfies both the continuous and discrete Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé V. In the end, by using Dyson's Coulomb fluid approach, we consider the large $n$ asymptotic behavior of our problem at the soft edge, which gives rise to the Painlevé XXXIV equation.

arXiv comment23 pages

References in corpus (7)

• Painleve IV and degenerate Gaussian Unitary Ensembles
• Painlevé VI and Hankel determinants for the generalized Jacobi Weight
• Asymptotics for a special solution of the thirty fourth Painleve equation
• Painlevé Transcendents and the Hankel Determinants Generated by a Discontinuous Gaussian Weight
• Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment, from Finite $n$ to Double Scaling
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Cited by in corpus (4)

• Differential, Difference and Asymptotic Relations for Pollaczek-Jacobi Type Orthogonal Polynomials and Their Hankel Determinants
• Painlevé V and the Hankel Determinant for a Singularly Perturbed Jacobi Weight
• Semi-classical Jacobi Polynomials, Hankel Determinants and Asymptotics
• Painlevé VI, Painlevé III and the Hankel Determinant Associated with a Degenerate Jacobi Unitary Ensemble