发表机构
City University of Hong Kong（香港城市大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对具有高阶边缘奇异性的酉随机矩阵系综，建立了Hankel行列式的大n渐近性，显式计算了常数项，并证明了特征值相关核的普适性，方法基于Deift-Zhou非线性最速下降分析。
AI 中文摘要
我们建立了具有高阶边缘奇异性的酉随机矩阵系综的Hankel行列式的大$n$渐近性。我们关注平衡测度支撑在单个区间上且极限特征值密度以$2k+\frac{1}{2}$阶（$k \in \mathbb{N}$）消失的系综。值得注意的是，我们显式地计算了渐近展开中的常数项，该常数项涉及与Painlevé I（$P_{\rm I}^{2k}$）层次相关的哈密顿量的正则化积分。作为副产品，我们还证明了在该奇异边缘附近特征值相关核的普适性，并推导出一个通过$P_{\rm I}^{2k}$方程的特殊解相关的函数表达的极限核。我们的方法依赖于正交多项式的Riemann-Hilbert问题的Deift-Zhou非线性最速下降分析。
英文摘要
We establish the large-$n$ asymptotics of Hankel determinants for unitary random matrix ensembles possessing a higher-order edge singularity. We focus on ensembles where the equilibrium measure is supported on a single interval and the limiting eigenvalue density vanishes to order $2k+\frac{1}{2}$ for $k \in \mathbb{N}$. Notably, we explicitly evaluate the constant term in the asymptotic expansion, which involves a regularized integral of the Hamiltonian associated with the Painlevé I ($P_{\rm I}^{2k}$) hierarchy. As a by-product, we also prove the universality of the eigenvalue correlation kernel near this singular edge and derive a limiting kernel expressed through functions related to a special solution of the $P_{\rm I}^{2k}$ equation. Our method relies on the Deift-Zhou nonlinear steepest descent analysis for the Riemann-Hilbert problem of orthogonal polynomials.
Comments47 pages, 5 figures
