On Expanding Neighborhoods of Local Universality of Gaussian Unitary Ensembles

被引:0
作者
M. A. Lapik
D. N. Tulyakov
机构
[1] Russian Academy of Sciences,Keldysh Institute of Applied Mathematics
来源
Proceedings of the Steklov Institute of Mathematics | 2018年 / 301卷
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摘要
The classical universality theorem states that the Christoffel–Darboux kernel of the Hermite polynomials scaled by a factor of 1/n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\sqrt n$$\end{document} tends to the sine kernel in local variables x˜,y˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde x,\tilde y$$\end{document} in a neighborhood of a point x∗∈(−2,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^*\in(-\sqrt 2,\sqrt 2)$$\end{document}). This classical result is well known for x~,y~∈K⋐R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde x,\tilde y\in{K}\Subset\mathbb{R}$$\end{document}. In this paper, we show that this classical result remains valid for expanding compact sets K = K(n). An interesting phenomenon of admissible dependence of the expansion rate of compact sets K(n) on x* is established. For x∗∈(−2,2)∖{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^*\in(-\sqrt 2,\sqrt 2)\backslash\left\{0\right\}$$\end{document}) and for x* = 0, there are different growth regimes of compact sets K(n). A transient regime is found.
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页码:170 / 179
页数:9
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