Noise sensitivity in continuum percolation
Noise sensitivity in continuum percolation
复制标题
连续渗滤中的噪声敏感性
DOI：
10.1007/s11856-014-1038-y
复制
发表时间：
2011
影响因子：
1
通讯作者：
R. Morris
中科院分区：
文献类型：
--
作者：
Daniel Ahlberg;Erik I. Broman;Simon Griffiths;R. Morris
我们证明泊松布尔模型（也称为吉尔伯特圆盘模型）在临界点对噪声敏感。这是连续渗流模型的第一个此类结果，也是第一个涉及临界概率 pc ≠ 1/2 的渗流模型的结果。我们的证明使用本杰明-卡莱-施拉姆定理的一个版本来测量有偏差的产品。凯勒和金德勒最近证明了这一结果的定量版本。我们从无偏版本中简单推导出非定量结果。我们还开发了一种相当通用的方法，通过 pc 远离零的离散模型来逼近连续渗流模型；该方法基于非均匀超图的极值结果。
We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first which involves a percolation model with critical probability pc ≠ 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with pc bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.