Gaussian Unitary Ensemble with Boundary Spectrum Singularity and σ‐Form of the Painlevé II Equation

Abstract

We consider the Gaussian unitary ensemble perturbed by a Fisher–Hartwig singularity simultaneously of both root type and jump type. In the critical regime where the singularity approaches the soft edge, namely, the edge of the support of the equilibrium measure for the Gaussian weight, the asymptotics of the Hankel determinant and the recurrence coefficients, for the orthogonal polynomials associated with the perturbed Gaussian weight, are obtained and expressed in terms of a family of smooth solutions to the Painlevé XXXIV equation and the σ‐form of the Painlevé II equation. In addition, we further obtain the double scaling limit of the distribution of the largest eigenvalue in a thinning procedure of the conditioning Gaussian unitary ensemble, and the double scaling limit of the correlation kernel for the critical perturbed Gaussian unitary ensemble. The asymptotic properties of the Painlevé XXXIV functions and the σ‐form of the Painlevé II equation are also studied.

On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation

“…and(1.19), the above integral is integrated toward +∞ instead of from −∞, due to the minus sign in the change of variable in(1.8). Moreover, unlike the tronquée solutions, the Hamiltonian H(s) is pole-free on the real line; see[50, Theorem 1]. Therefore, no Cauchy principal value is needed in the definition of I 2 for all ω ≥ 0.Our second result is the following theorem.…”

“…As shown in [39] (see also [23,Chapter 11]), these solutions belong to the classical tronquée solutions due to Boutroux [10], which means they are pole-free near infinity in one or more sectors of opening angle 2π/3 of the complex plane. Their applications in random matrix theory and asymptotics of orthogonal polynomials can be found in [12,50].…”

“…As the parameter varies, we indeed find such kind of transition from the non-trivial constant term therein. More precisely, we start with a fact that the limiting kernel in [11] can be viewed as a special case of the general Painlevé XXXIV kernel K P 34 α,ω (x, y; t) with the parameters α > − 1 2 , ω = 0, where t comes from the limit c 1 n 2/3 (1 − n N ) → t for some constant c 1 ; see [32,50]. For α > − 1 2 , ω 0 and t ∈ R, let K P 34 α,ω | [s,+∞) be the trace-class operator acting on L 2 (s, +∞) with the Painlevé XXXIV kernel K P 34 α,ω (x, y; t).…”

“…Proof. The existence of unique solution to the above RH problem is proved in [50, Proposition 1] and the relation (2.10) is given in [50,Equation (44)], which also implies that H(s) is pole-free for s ∈ R.…”

“…The detailed nonlinear steepest descend analysis of the RH problem for Ψ as s → −∞ was performed in [33] for α > −1/2 and ω > 0, and by Wu, Xu and Zhao in [50] for the case α > −1/2 and ω = 0. Again, we give a sketch of the analysis in this section for later use.…”