We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek–Jacobi type weight wPJ2(x,t; α,β) = xα(1 − x)βe−t x(1−x), where t ∈ [0,∞), α > 0, β > 0 and 0 < x < 1. Based on our observation, we find that this weight includes the symmetric constraint wPJ2(x,t; α,β) = wPJ2(1 − x,t; β,α). Our main results obtained here include two aspects: (1) Strong asymptotics: we deduce strong asymptotics of monic orthogonal polynomials with respect to the above weight function in different regions in the complex plane when the polynomial degree n goes to infinity. Because of the effect of t x(1−x) for varying t, the asymptotic behavior in a neighborhood of point 1 is described in terms of the Airy function as ζ = 2n2t →∞,n →∞, but the Bessel function as ζ → 0,n →∞. Due to symmetry, the similar local asymptotic behavior near the singular point x = 0 can be derived. (2) Limiting eigenvalue correlation kernels: We calculate the limit of the eigenvalue correlation kernel of the corresponding unitary random matrix ensemble in the bulk of the spectrum described by the sine kernel, and at both sides of hard edge, expressed as a Painlevé III kernel. Our analysis is based on the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.

中文翻译：

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奇异扰动的 Pollaczek-Jacobi 型酉系综中的临界边缘行为

我们研究与奇异扰动的 Pollaczek-Jacobi 型权重相关的正交多项式 wPJ2(X,吨; α,β) = Xα(1 - X)βe-吨 X(1-X), 在哪里吨 ∈ [0,∞),α > 0,β > 0和0 < X < 1. 根据我们的观察，我们发现这个权重包括对称约束w磷Ĵ2(X,吨; α,β) = w磷Ĵ2(1 - X,吨; β,α). 我们这里得到的主要结果包括两个方面： (1) 强渐近性：我们在多项式度数的情况下，在复平面的不同区域对上述权函数推导出一元正交多项式的强渐近性。n走向无穷大。因为效果 吨 X(1-X)对于不同的吨, 点邻域的渐近行为1用艾里函数描述为ζ = 2n2吨 →∞,n →∞，但贝塞尔函数为ζ → 0,n →∞. 由于对称性，奇异点附近的类似局部渐近行为X = 0可以导出。(2) 限制特征值相关核：我们计算对应酉随机矩阵集合的特征值相关核在正弦核描述的大部分频谱中的限制，并且在硬边缘的两侧，表示为 Painlevé III 核. 我们的分析基于黎曼-希尔伯特问题的 Deift-Zhou 非线性最速下降法。