On questions of uniqueness for the vacant set of Wiener sausages and Brownian interlacements

• 成果类型: Article
• 署名作者: Mu, Yingxin; Sapozhnikov, Artem
• 署名单位: Leipzig University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01315-y
• 发表日期: 2024
• 页码: 703-751
• 关键词: percolation; times

摘要

We consider connectivity properties of the vacant set of (random) ensembles of Wiener sausages in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}<^>d$$\end{document} in the transient dimensions d >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \ge 3$$\end{document}. We prove that the vacant set of Brownian interlacements contains at most one infinite connected component almost surely. For finite ensembles of Wiener sausages, we provide sharp polynomial bounds on the probability that their vacant set contains at least 2 connected components in microscopic balls. The main proof ingredient is a sharp polynomial bound on the probability that several Brownian motions visit jointly all hemiballs of the unit ball while avoiding a slightly smaller ball.