$\mathbb{Z}^{d}$上从紧凸子集到无限的第一遍渗流的最大流

IF 2.1 1区数学 Q1 STATISTICS & PROBABILITY

Annals of Probability Pub Date : 2020-03-01 DOI:10.1214/19-aop1367

Barbara Dembin

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引用次数: 0

摘要

我们考虑Z^d上的标准第一通道渗流模型，其分布G在R+上允许指数矩。研究了R^d的紧凸子集a与无穷远之间的极大流。最大流量的研究是与最小化容量边集的研究相联系的，这些边集从无穷远切割出A。我们证明了在nA和无穷之间φ(nA)/n^ (d - 1)的重标最大流几乎肯定会收敛于一个依赖于a的确定性常数。这个常数对应于边界∂a (a)的容量，并且是一个确定性函数在∂a上的积分。这一结果在维度2中得到了显示，并由Garet在[6]中对更高维度进行了推测。

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The maximal flow from a compact convex subset to infinity in first passage percolation on $\mathbb{Z}^{d}$

We consider the standard first passage percolation model on Z^d with a distribution G on R+ that admits an exponential moment. We study the maximal flow between a compact convex subset A of R^d and infinity. The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut A from infinity. We prove that the rescaled maximal flow between nA and infinity φ(nA)/n^ (d−1) almost surely converges towards a deterministic constant depending on A. This constant corresponds to the capacity of the boundary ∂A of A and is the integral of a deterministic function over ∂A. This result was shown in dimension 2 and conjectured for higher dimensions by Garet in [6].

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来源期刊

Annals of Probability 数学-统计学与概率论

CiteScore

4.60

自引率

8.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The Annals of Probability publishes research papers in modern probability theory, its relations to other areas of mathematics, and its applications in the physical and biological sciences. Emphasis is on importance, interest, and originality – formal novelty and correctness are not sufficient for publication. The Annals will also publish authoritative review papers and surveys of areas in vigorous development.