Laguerre unitary ensembles with jump discontinuities, PDEs and the coupled Painlevé V system

We study the Hankel determinant generated by the Laguerre weight with jump discontinuities at tk, k = 1, ... , m. By employing the ladder operator approach to establish Riccati equations, we show that sigma n(t1, ... , tm), the logarithmic derivative of the n-dimensional Hankel determinant, satisfies an m-variable generalization of the sigma-form of a Painleve V equation. Through investigating the Riemann- Hilbert problem for the associated orthogonal polynomials and via the Lax pair, we express sigma n in terms of solutions of a coupled Painleve V system. We also build relations between the auxiliary quantities introduced in the above two methods, which provides connections between the Riccati equations and the Lax pair. In addition, when each tk tends to the hard edge of the spectrum and n goes to infinity, the scaled sigma n is shown to satisfy a generalized sigma-form of a Painleve III equation.(c) 2023 Elsevier B.V. All rights reserved.