Painlevé III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight

In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight truerightleftwpJ(x,t)=e−txxαfalse(1−xfalse)β,1emt≥0,rightleft1emα>0,1emβ>0,1emx∈false[0,1false].The uniform asymptotic expansions for the monic orthogonal polynomials on the interval (0,1) and outside this interval are obtained. Moreover, near x=0, the uniform asymptotic expansion involves Airy function as ς=2n2t→∞,n→∞, and Bessel function of order α as ς=2n2t→0,n→∞; in the neighborhood of x=1, the uniform asymptotic expansion is associated with Bessel function of order β as n→∞. The recurrence coefficients and leading coefficient of the orthogonal polynomials are expressed in terms of a particular Painlevé III transcendent. We also obtain the limit of the kernel in the bulk of the spectrum. The double scaled logarithmic derivative of the Hankel determinant satisfies a σ‐form Painlevé III equation. The asymptotic analysis is based on the Deift and Zhou's steepest descent method.

“…

Ψ false(ζ, s false) = Q false(s false) (I + scriptO false(ζ false)) e^{\frac{s}{ζ} σ_{3}} ζ^{\frac{α}{2} σ_{3}} \{\begin{matrix} left I, & left ζ \in Ω_{1} ⋃ Ω_{4}, \\ left ((), \begin{matrix} 1 & 0 \\ - e^{α π i} & 1 \end{matrix}), & left ζ \in Ω_{2}, \\ left ((), \begin{matrix} 1 & 0 \\ e^{- α π i} & 1 \end{matrix}), & left ζ \in Ω_{3}, \end{matrix}

with the argument

arg ζ \in (- π, π)

and

Q (s)

given by

Q false(s false) = \sqrt{\frac{1 + q^{'} (s)}{2}} (\begin{matrix} 1 & \frac{i r (s)}{1 + q^{'} false(s false)} \\ \frac{i (1 - q^{'} false(s false))}{r^{'} false(s false)} & 1 \end{matrix}) C^{σ_{3}},

with C a nonzero constant, which can be found in, see (4.4) therein. Moreover,

r (s)

satisfies the boundary conditions for

α > 0

r false(s false) = ...

…”

Section: Deift–zhou Steepest Descent Analysismentioning

confidence: 99%

“…Although its explicit solution of this model cannot write down for finite

ς = 2 n^{2} t, Ψ false(ξ, ς false)

can be approximated by the Bessel model RHp as

ς \to 0

and the Airy model RHp as

ς \to \infty

, separately. For the transition between these two regimes, see . We state main results as follows.…”

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials

Chen

Fan

2019

“…consider the asymptotics of the partition function associated with the Gaussian weight perturbed by an essential singularity and they get Painlevé III‐type asymptotics. In and , Xu et al. also obtain Painlevé III‐type asymptotics of the Hankel determinants associated with the Laguerre weight with an essential singularity at the hard edge.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 96%

Painlevé III Asymptotics of Hankel Determinants for a Perturbed Jacobi Weight

Zeng

Zhao

2015

Self Cite

We study the Hankel determinants associated with the weight w(x;t)=(1−x2)β(t2−x2)αh(x),x∈(−1,1),where β>−1, α+β>−1, t>1, h(x) is analytic in a domain containing [ − 1, 1] and h(x)>0 for x∈[−1,1]. In this paper, based on the Deift–Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as n→∞ and t→1. We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo–Miwa–Okamoto σ‐function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained.

“…In , asymptotic formulas have been derived of the Hankel determinants associated with the Jacobi weight perturbed by a Fisher–Hartwig singularity close to the hard edge

x = 1

, involving the Jimbo–Miwa–Okamoto σ‐form of the Painlevé III equation. The Painlevé equations play an important role in the asymptotic study of the Hankel determinants; see , , , and .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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

Wu¹,

Zhao³

2017

Self Cite

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.