Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System

Abstract

We study the Hankel determinant generated by the Laguerre weight with jump discontinuities at $t_k,k=1,\cdots ,m$. By employing the ladder operator approach to establish Riccati equations, we show that ${\sigma }_n(t_1,\cdots ,t_m)$, the logarithmic derivative of the $n$-dimensional Hankel determinant, satisfies a generalization of the $\sigma$-from of Painlevé V equation. Through investigating the Riemann-Hilbert problem for the associated orthogonal polynomials and via Lax pair, we express ${\sigma }_n$ in terms of solutions of a coupled Painlevé V system. We also build relations between the auxiliary quantities introduced in the above two methods, which provides connections between the Riccati equations and Lax pair. In addition, when each $t_k$ tends to the hard edge of the spectrum and $n$ goes to $\infty$, the scaled ${\sigma }_n$ is shown to satisfy a generalized Painlevé III equation.

References in corpus (12)

• Fredholm Determinants, Differential Equations and Matrix Models
• Level-Spacing Distributions and the Bessel Kernel
• Asymptotics of Hankel determinants with a one-cut regular potential and Fisher-Hartwig singularities
• Random Matrix Models, Double-Time Painlevé Equations, and Wireless Relaying
• Large gap asymptotics for Airy kernel determinants with discontinuities
• Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles
• 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
• Tracy-Widom distributions in critical unitary random matrix ensembles and the coupled Painlevé II system
• Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems
• Gaussian unitary ensembles with pole singularities near the soft edge and a system of coupled Painlevé XXXIV equations
• On the ratio probability of the smallest eigenvalues in the Laguerre Unitary Ensemble

Cited by in corpus (1)

• Exponential moments for disk counting statistics of random normal matrices in the critical regime