On the size of a finite vacant cluster of random interlacements with small intensity

• 成果类型: Article
• 署名作者: Teixeira, Augusto
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0283-x
• 发表日期: 2011
• 页码: 529-574
• 关键词: discrete cylinder; disconnection time; random-walk; set

摘要

In this paper, we establish some properties of percolation for the vacant set of random interlacements, for d a parts per thousand yen 5 and small intensity u. The model of random interlacements was first introduced by Sznitman in (Ann Math, arXiv:0704.2560, 2010). It is known that, for small u, almost surely there is a unique infinite connected component in the vacant set left by the random interlacements at level u, see Sidoravicius and Sznitman (Commun Pure Appl Math 62(6):831-858, 2009) and Teixeira (Ann Appl Probab 19(1):454-466, 2009). We estimate here the distribution of the diameter and the volume of the vacant component at level u containing the origin, given that it is finite. This comes as a by-product of our main theorem, which proves a stretched exponential bound on the probability that the interlacement set separates two macroscopic connected sets in a large cube. As another application, we show that with high probability, the unique infinite connected component of the vacant set is ubiquitous in large neighborhoods of the origin.