Spherical integrals of sublinear rank

• 成果类型: Article
• 署名作者: Husson, Jonathan; Ko, Justin
• 署名单位: University of Michigan System; University of Michigan; Ecole Normale Superieure de Lyon (ENS de LYON)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01402-8
• 发表日期: 2025
• 页码: 1-88
• 关键词: CENTRAL LIMIT-THEOREMS; free-energy; LARGEST EIGENVALUE; large deviations; fundamental limits; horns problem; spin; MODEL; asymptotics; INFORMATION

摘要

We consider the asymptotics of rank k spherical integrals when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = o(N)$$\end{document}. We prove that the sublinear rank spherical integrals are approximately the products of rank 1 spherical integrals. Our formulas extend the results for rank k spherical integrals proved by Guionnet and Ma & iuml;da in Guionnet and Ma & iuml;da (J. Funct. Anal. 222:435-490, 2005) and Husson and Guionnet in Guionnet and Husson (ALEA. 19:769-797, 2022) which are only valid for k finite and independent of N. These approximations will be used to prove a large deviation principle for the joint 2k(N) extreme eigenvalues for sharp sub-Gaussian Wigner matrices and for additive deformations of GOE/GUE matrices. Furthermore, our results will be used to compute the free energies of spherical SK vector spin glasses and the mutual information for matrix estimation problems when the dimensions of the spins or signals have sublinear growth.