On differential systems related to generalized Meixner and deformed Laguerre orthogonal polynomials

In this paper we present a connection between systems of differential equations for the recurrence coefficients of polynomials orthogonal with respect to the generalized Meixner and the deformed Laguerre weights. It is well-known that the recurrence coefficients of both generalized Meixner and deformed Laguerre orthogonal polynomials can be expressed in terms of solutions of the fifth Painlevé equation but no explicit relation between systems of differential equations for the recurrence coefficients was known. We also present certain limits in which the recurrence coefficients can be expressed in terms of solutions of the Painlevé XXXIV equation, which in the deformed Laguerre case extends previous studies and in the generalized Meixner case is a new result

Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge

We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support. Their behavior is described in terms of the Ablowitz-Segur family of solutions to the Painlev\'e II equation. Our results complement the ones in [Xu,Zhao,2011]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painlev\'e II transcendents, and we also prove a new result on the poles of the Ablowitz-Segur solutions to the Painlev\'e II equation. We also highlight applications of our results in random matrix theory.Comment: 35 pages, 4 figure

Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials

AbstractRecurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function w such that w′w is a rational function) are shown to be solutions of nonlinear differential equations with respect to a well-chosen parameter, according to principles established by D. Chudnovsky and G. Chudnovsky. Examples are given. For instance, the recurrence coefficients in an + 1Pn + 1 (x) = xpn(x) − anpn − 1 (x) of the orthogonal polynomials related to the weight exp (− x44 − tx2) on R satisfy 4an3än = (3an4 + 2tan2 − n)(an4 + 2tan2 + n), and an2 satisfies a Painlevé PIV equation

Hamiltonian structure for a differential system from a modified Laguerre weight via the geometry of the modified third Painlevé equation

Recurrence coefficients of semi-classical orthogonal polynomials are often related to the solutions of special nonlinear second-order differential equations known as the Painlevé equations. Each Painlevé equation can be written in a standard form as a non-autonomous Hamiltonian system, so it is natural to ask whether differential systems satisfied by the recurrence coefficients also possess Hamiltonian structures. We consider recurrence coefficients for a modified Laguerre weight which satisfy a differential system known to be related to the modified third Painlevé equation and identify a Hamiltonian structure for it by constructing its space of initial conditions. We also discuss a transformation from this system to the modified third Painlevé equation which simultaneously identifies a discrete system for the recurrence coefficients with a discrete Painlevé equation

Painlev\'e Functions in Statistical Physics

We review recent progress in limit laws for the one-dimensional asymmetric simple exclusion process (ASEP) on the integer lattice. The limit laws are expressed in terms of a certain Painlev\'e II function. Furthermore, we take this opportunity to give a brief survey of the appearance of Painlev\'e functions in statistical physics.Comment: Revised version updates some reference