Sparse reconstruction in spin systems II: Ising and other factor of IID measures

• 成果类型: Article; Early Access
• 署名作者: Galicza, Pal; Pete, Gabor
• 署名单位: HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics; Budapest University of Technology & Economics
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01395-4
• 发表日期: 2025
• 关键词: glauber dynamics; noise sensitivity; finitary codings; PHASE-TRANSITION; percolation; magnetization; THEOREMS; MODEL; inequalities; entropy

摘要

For a sequence of Boolean functions fn:{-1,1}Vn ->{-1,1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_n : \{-1,1\}<^>{V_n} \longrightarrow \{-1,1\}$$\end{document}, with random input given by some probability measure Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}_n$$\end{document}, we say that there is sparse reconstruction for fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_n$$\end{document} if there is a sequence of subsets Un subset of Vn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_n \subseteq V_n$$\end{document} of coordinates satisfying |Un|=o(|Vn|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|U_n| = o(|V_n|)$$\end{document} such that knowing the spins in Un\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_n$$\end{document} gives us a non-vanishing amount of information about the value of fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_n$$\end{document}. In the first part of this work, we showed that if the Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}_n$$\end{document}s are product measures, then no sparse reconstruction is possible for any sequence of transitive functions. In this sequel, we consider spin systems that are relatives of IID measures in one way or another, with our main focus being on the Ising model on finite transitive graphs or exhaustions of lattices. We prove that no sparse reconstruction is possible for the entire high temperature regime on Euclidean boxes and the Curie-Weiss model, while sparse reconstruction for the majority function of the spins is possible in the critical and low temperature regimes. We give quantitative bounds for two-dimensional boxes and the Curie-Weiss model, sharp in the latter case. The proofs employ several different methods, including factor of IID and FK random cluster representations, strong spatial mixing, a generalization of discrete Fourier analysis to Divide-and-Color models, and entropy inequalities.