Random Fields on Model Sets with Localized Dependency and Their Diffraction
Random Fields on Model Sets with Localized Dependency and Their Diffraction
复制标题
具有局域依赖性的模型集上的随机场及其衍射
DOI：
10.1007/s10955-012-0588-5
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发表时间：
2012
影响因子：
1.6
通讯作者：
Yohji Akama and Shinji Iizuka
中科院分区：
文献类型：
--
作者：
Taisuke Matsuno;Hisashi Naito;Sunpei Hitotsugi;Sota Sato;Motoko Kotani and Hiroyuki Isobe;小谷 元子;Tadao Oda;Yohji Akama and Shinji Iizuka
对于一般离散集上的随机场，我们引入了一个条件，即每个站点的相关范围在预定义的紧致集D内。对于定义在满足自然几何条件的模型集Λ上的随机场ω，我们给出了一种计算随机场衍射测度的方法.该方法将随机场划分为有限个随机场，每个随机场都是独立的，并允许大数定律。ω的衍射测度几乎必然由一个纯点分量和一个绝对连续分量组成。前者是期望E[ω]的衍射测度，而ω的绝对连续分量的逆傅里叶变换则是满足一个简单公式的加权Dirac梳。此外，如果权在Λ的内部空间上是连续的，则纯点分量将在简单的精确公式中定量地理解。然后给出了模型集上随机场的衍射测度仍为纯点的一个充分条件。
For a random field on a general discrete set, we introduce a condition that the range of the correlation from each site is within a predefined compact setD. For such a random fieldωdefined on the model setΛthat satisfies a natural geometric condition, we develop a method to calculate the diffraction measure of the random field. The method partitions the random field into a finite number of random fields, each being independent and admitting the law of large numbers. The diffraction measure ofωconsists almost surely of a pure-point component and an absolutely continuous component. The former is the diffraction measure of the expectation E[ω], while the inverse Fourier transform of the absolutely continuous component ofωturns out to be a weighted Dirac comb which satisfies a simple formula. Moreover, the pure-point component will be understood quantitatively in a simple exact formula if the weights are continuous over the internal space ofΛ. Then we provide a sufficient condition that the diffraction measure of a random field on a model set is still pure-point.