First Passage Percolation Has Sublinear Distance Variance

Itai Benjamini; Gil Kalai; Oded Schramm

First Passage Percolation Has Sublinear Distance Variance

概率论 2008-11-26 v5 数学物理 math.MP

作者: Itai Benjamini , Gil Kalai , Oded Schramm

摘要

Let $0<a<b<\infty$ , and for each edge $e$ of $Z^d$ let $\omega_e=a$ or $\omega_e=b$ , each with probability 1/2, independently. This induces a random metric $\dist_\omega$ on the vertices of $Z^d$ , called first passage percolation. We prove that for $d>1$ the distance $dist_\omega(0,v)$ from the origin to a vertex $v$ , $|v|>2$ , has variance bounded by $C |v|/\log|v|$ , where $C=C(a,b,d)$ is a constant which may only depend on $a$ , $b$ and $d$ . Some related variants are also discussed

关键词

percolation theory distance sets random walks

引用

@article{arxiv.math/0203262,
  title  = {First Passage Percolation Has Sublinear Distance Variance},
  author = {Itai Benjamini and Gil Kalai and Oded Schramm},
  journal= {arXiv preprint arXiv:math/0203262},
  year   = {2008}
}

备注

Replaced theorem 2 (which was incorrect) by a new theorem

First Passage Percolation Has Sublinear Distance Variance

摘要

关键词

引用

备注

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