092 Yang Chen

Laguerre Unitary Ensembles with Multiple Discontinuities, PDE, and the Coupled Painlevé V System // Abstract:// We study the Hankl generated by the Laguerre weight with jump discontinuities at t_k, k=1,2,..,m. By employing the ladder operator approach we establish (multi-time) Riccati equations, to show that sigma_n(t_1, ...,t_m), the log derivative of the nxn Hankel determinant, satisfies a generalization of the sigma of a Painlevé V equation. Through investigating the Riemann-Hibert problem (or Homogenous Hilbert Problem) for the orthogonal polynomials generated by the LUEMD 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 provide connections between the Riccati equations and the Lax Pair. // In addition, when each t_k tends to the hard edge of the spectrum and n goes to infinity,