论文标题
奇异的边缘行为在奇异的扰动的Pollaczek-Jacobi型统一合奏中
Critical edge behavior in the singularly perturbed Pollaczek-Jacobi type unitary ensemble
论文作者
论文摘要
在本文中，我们研究了与奇异扰动的Pollaczek-Jacobi类型重量$$ W_ {P_J2}（X，T）= E^{ - \ frac {t} {x（1-x（1-x）$ ge 0- $ there $ there $ there $ x^α（$ ther） $α> 0 $，$β> 0 $和$ x \ in [0,1]。$我们在此获得的主要结果包括两个方面： {I.强渐近学：}我们以不同的间隔$（0,1）$（0,1）$（0,1）$（0,1）$（0,1）和间隔$ \ MATHBB {C} \ MATHBB {C} \ BACKSLASH（0,1）$以不同的间隔$（0,1）$（0,1）$获得强的渐近膨胀;由于$ \ frac {t} {x（1-x）} $的效果对于不同的$ t $，因此在硬边缘$ 0 $上的不同渐近行为和$ 1 $的不同渐近方案。具体而言，均匀的渐近行为可以作为$ 1 $的附近的通风函数表示为$ζ= 2n^2t \ to \ infty，n \ to \ infty $，而Bessel函数则以$ζ\ \ to \ 0，n \至n \ to \ suftty $。 {II。通用性：}我们分别计算大部分频谱中的特征值相关核的极限，并且在硬边的两侧，这将涉及与特定的paraclev $ \ actute {e} $ \ a} $ \ uppercase \ uppercase \ uppercase \ uppercase \ uppercase \ upperfer \ uplover {此外，我们还证明，贝塞尔内核可以将$ψ$ - funcation近似为$ζ\ to $ζ\至0 $，而与通风的内核则为$ζ\至\ fos infty $。我们的分析基于Riemann-Hilbert问题的Deift-Zhou非线性陡峭下降方法。
In this paper, we study the strong asymptotic for the orthogonal polynomials and universality associated with singularly perturbed Pollaczek-Jacobi type weight $$w_{p_J2}(x,t)=e^{-\frac{t}{x(1-x)}}x^α(1-x)^β, $$ where $t \ge 0$, $α>0$, $β>0$ and $x \in [0,1].$ Our main results obtained here include two aspects: { I. Strong asymptotics:} We obtain the strong asymptotic expansions for the monic Pollaczek-Jacobi type orthogonal polynomials in different interval $(0,1)$ and outside of interval $\mathbb{C}\backslash (0,1)$, respectively; Due to the effect of $\frac{t}{x(1-x)}$ for varying $t$, different asymptotic behaviors at the hard edge $0$ and $1$ were found with different scaling schemes. Specifically, the uniform asymptotic behavior can be expressed as a Airy function in the neighborhood of point $1$ as $ζ= 2n^2t \to \infty, n\to \infty$, while it is given by a Bessel function as $ζ\to 0, n \to \infty$. { II. Universality:} We respectively calculate the limit of the eigenvalue correlation kernel in the bulk of the spectrum and at the both side of hard edge, which will involve a $ψ$-functions associated with a particular Painlev$\acute{e}$ \uppercase\expandafter{\romannumeral3} equation near $x=\pm 1$. Further, we also prove the $ψ$-funcation can be approximated by a Bessel kernel as $ζ\to 0$ compared with a Airy kernel as $ζ\to \infty$. Our analysis is based on the Deift-Zhou nonlinear steepest descent method for the Riemann-Hilbert problems.
