Fluctuations in the logarithmic energy for zeros of random polynomials on the sphere

• 成果类型: Article
• 署名作者: Michelen, Marcus; Yakir, Oren
• 署名单位: University of Illinois System; University of Illinois Chicago; University of Illinois Chicago Hospital; Tel Aviv University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01334-9
• 发表日期: 2025
• 页码: 569-626
• 关键词: random complex zeros; statistics; points; riesz

摘要

Smale's Seventh Problem asks for an efficient algorithm to generate a configuration of n points on the sphere that nearly minimizes the logarithmic energy. As a candidate starting configuration for this problem, Armentano, Beltr & aacute;n and Shub considered the set of points given by the stereographic projection of the roots of the random elliptic polynomial of degree n and computed the expected logarithmic energy. We study the fluctuations of the logarithmic energy associated to this random configuration and prove a central limit theorem. Our approach shows that all cumulants of the logarithmic energy are asymptotically linear in n, and hence the energy is well-concentrated on the scale of n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{n}$$\end{document}.