Limit theorems for maximum flows on a lattice

• 成果类型: Article
• 署名作者: Zhang, Yu
• 署名单位: University of Colorado System; University of Colorado at Colorado Springs
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0775-z
• 发表日期: 2018
• 页码: 149-202
• 关键词: 1st-passage percolation; SURFACES

摘要

We independently assign a non-negative value, as a capacity for the quantity of flows per unit time, with a distribution F to each edge on the lattice. We consider the maximum flows through the edges from a source to a sink in a large cube. In this paper, we show that the ratio of the maximum flow and the size of the source is asymptotic to a constant. This constant is denoted by the flow constant. By the max-flow and min-cut theorem, this is equivalent to a statement about the asymptotic behavior of the minimal value assigned to any surface on the large cube. We can also show that there exists such a surface that is proportional to the size of the faces of the cube.