Painlevé V and the distribution function of discontinuous linear statistics in the Laguerre Unitary Ensembles

2008/07/29 by Estelle Basor, Yang Chen, Basor, Estelle +1
Mathematics · Physics and Astronomy · #42C05 #82B #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Random Matrices and Applications #Statistical Distribution Estimation and Applications #math-ph #math.MP #msc:42C05 #msc:82B

paper · pdf · doi:10.48550/arxiv.0807.4758

openalex publication_date 2008/07/29 · arxiv created 2008/11/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study the characteristic or generating function of a certain discontinuous linear statistics of the Laguerre unitary ensembles and show that this is a particular fifth Painléve transcendant in the variable t, the position of the discontinuity. The proof of the ladder operators adapted to orthogonal polynomial with discontinuous weight announced sometime ago is presented here, followed by the non-linear difference equations satisfied by two auxiliary quantities and the derivation of the Painléve equation.

Citations

• Toeplitz matrices, translation kernels and a related problem in probability theory
• A differential equation for orthogonal polynomials
• Freud equations for Legendre polynomials on a circular arc and solution of the Grünbaum–Delsarte–Janssen–Vries problem
• Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II

Related

• Differential equations for deformed Laguerre polynomials
• Painléve III and a singular linear statistics in Hermitian random matrix ensembles I
• Painlevé transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles
• Painlevé V for a Jacobi unitary ensemble with random singularities
• Application of the τ-function theory of Painlevé equations to random matrices: \PV, \PIII, the LUE, JUE and CUE