Local central limit theorem for gradient field models

• 成果类型: Article
• 署名作者: Wu, Wei
• 署名单位: New York University; NYU Shanghai
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01330-z
• 发表日期: 2025
• 页码: 1-40
• 关键词: scaling limit

摘要

We consider the gradient field model in -N,N2 boolean AND Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ -N,N\right] <^>{2}\cap {\mathbb {Z}}<^>{2}$$\end{document} with a uniformly convex interaction potential. Naddaf-Spencer (Comm Math Phys 183(1):55-84, 1997) and Miller (Comm Math Phys 908(3):591-639, 2011) proved that the macroscopic averages of linear statistics of the field converge to a continuum Gaussian free field. In this paper we prove the distribution of phi(0)/logN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (0)/\sqrt{\log N}$$\end{document} converges uniformly in R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document} to a Gaussian density, with a Berry-Esseen type bound. This implies the distribution of phi(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (0)$$\end{document} is sufficiently 'Gaussian like' between [-logN,logN]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-\sqrt{\log N}, \sqrt{\log N}]$$\end{document}.