Critical edge behavior in the modified Jacobi ensemble and Painlevé equations

Abstract

We study the Jacobi unitary ensemble perturbed by an algebraic singularity at t > 1. For fixed t, this is the modified Jacobi ensemble studied by Kuijlaars et al. The main focus here, however, is the case when the algebraic singularity approaches the hard edge, namely t → 1 + .In the double scaling limit case when t − 1 is of the order of magnitude of 1/n 2 , n being the size of the matrix, the eigenvalue correlation kernel is shown to have a new limiting kernel at the hard edge 1, described by the ψ-functions for a certain second-order nonlinear equation. The equation is related to the Painlevé III equation by a Möbius transformation. It also furnishes a generalization of the Painlevé V equation, and can be reduced to a particular Painlevé V equation via the Bäcklund transformations in special cases. The transitions of the limiting kernel to Bessel kernels are also investigated, with n 2 (t − 1) being large or small.In the present paper, the approach is based on the Deift-Zhou nonlinear steepest descent analysis for Riemann-Hilbert problems.

Random Matrices with Merging Singularities and the Painlevé V Equation

“…In addition, we require zero to be contained in the interior of the support of µ V , i.e., a < 0 < b. If zero were outside of the support of µ V , no eigenvalues are expected near the origin for large n, and the singularities of the weight will not play a significant role for large n. The case where 0 is an edge point of the support of V is also interesting, but will not be studied here -we refer to [22] for results about the local eigenvalue correlations when a singularity lies near a soft edge, and to [33] when a singularity approaches a hard edge in a modified Jacobi ensemble.…”

“…This observation can be used to obtain information about extreme values of |P n (x)| for large n. This problem was investigated in [20], and in this context the authors needed large n asymptotics for integrals of the form (see in particular [20, Section 2]) 33) with θ ∈ [0, 1], ρ strictly positive and continuous on (−θ, θ), and α > 0. The authors note that one can differentiate (1.31) for k = 1 with respect to α and evaluate at α = 0 to obtain…”

Painlevé III Asymptotics of Hankel Determinants for a Perturbed Jacobi Weight

“…In the preceding papers , the authors have studied the transition asymptotics of the eigenvalue correlation kernel for the perturbed Jacobi unitary ensemble defined by the perturbed Jacobi weight given in , varying from the Bessel kernel to as the parameter t varies in (1, d ] for a fixed . A new class of universal behavior at the edge of the spectrum for the modified Jacobi ensemble is obtained and described in terms of the generalized Painlevé V equation, which in this case is equivalent to the Painlevé III equation after a Möbius transformation.…”

“…The identities are the starting points of our analysis in later sections. In Section , we outline the notations and formulas resulted from the RH analysis, obtained previously by the authors in . The proofs of Theorems – are provided in the last section, Section .…”

“…In , to construct the local parametrix in the nonlinear steepest descent analysis of the RH problems, Xu and Zhao introduce a modified version of the Painlevé V equation that is equivalent to the Painlevé III equation after a Möbius transformation. Proposition (Xu and Zhao ) Assume that solves where α and β are constants.…”