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Yu-Qiu Zhao

Publications and source records attributed to Yu-Qiu Zhao.

25 records · Page 2Linked to original sources
arXiv2015

Critical edge behavior in the modified Jacobi ensemble and Painlevé equations

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 {\it{et al.}} The main focus here, however, is the case when the algebraic singularity approaches the hard edge, namely $t\to 1^+$. 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.

math-ph↗
arXiv2014

Weights with both absolutely continuous and discrete components: Asymptotics via the Riemann-Hilbert approach

We study the uniform asymptotics for the orthogonal polynomials with respect to weights composed of both absolutely continuous measure and discrete measure, by taking a special class of the sieved Pollazek Polynomials as an example. The Plancherel-Rotach type asymptotics of the sieved Pollazek Polynomials are obtained in the whole complex plane. The Riemann-Hilbert method is applied to derive the results. A main feature of the treatment is the appearance of a new band consisting of two adjacent intervals, one of which is a portion of the support of the absolutely continuous measure, the other is the discrete band.

math.CV↗
arXiv2014

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

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight $w(x)=x^αe^{-x-t/x}$, $x\in (0, \infty)$, $t>0$ and $α>0$. When the matrix size $n\to\infty$, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for $t\in (0, d]$, $d>0$ fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large-$n$ asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift-Zhou nonlinear steepest descent method.

math.CA↗
arXiv2014

Uniform asymptotics for discrete orthogonal polynomials on infinite nodes with an accumulation point

In this paper, we develop the Riemann-Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an accumulation point. To illustrate our method, we consider the Tricomi-Carlitz polynomials $f_n^{(α)}(z)$ where $α$ is a positive parameter. Uniform Plancherel-Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [{\it SIAM J.\;Math.\;Anal} {\bf 25} (1994)] and [{{\it Proc.\;Amer.\;Math.\;Soc.\,}{\bf138} (2010)}].

math.CA↗
arXiv2014

Critical edge behavior and the Bessel to Airy transition in the singularly perturbed Laguerre unitary ensemble

In this paper, we study the singularly perturbed Laguerre unitary ensemble $$ \frac{1}{Z_n} (\det M)^αe^{- \textrm{tr}\, V_t(M)}dM, \qquad α>0, $$ with $V_t(x) = x + t/x$, $x\in (0,+\infty)$ and $t>0$. Due to the effect of $t/x$ for varying $t$, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves $ψ$-functions associated with a special solution to a new third-order nonlinear differential equation, which is then shown equivalent to a particular Painlevé III equation. The transition of this limiting kernel to the Bessel and Airy kernels is also studied when the parameter $t$ changes in a finite interval $(0, d]$. Our approach is based on Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems.

math-ph↗