局部依赖模型集上的随机场及其绕射
Random fields on model sets with localized dependency and their diffraction
摘要：
对于一般离散集上的随机场，我们引入了一个条件，即每个点的相关范围在预定紧集D内。对于定义在满足自然几何条件的模型集λ上的这样一个随机场Ω，我们给出了计算该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，每个随机场独立且满足大数定律。Ω的衍射测度几乎肯定由纯点分量和绝对连续分量组成。前者是期望E[omega]的衍射测度，而omega的绝对连续分量的傅里叶逆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，则纯点分量可以用一个简单的精确公式定量地理解。然后，我们给出了模型集上随机场的衍射测度仍是纯点的一个充分条件。
英文摘要：
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 set D. For such a random field omega defined on the model set Lambda that 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 of omega consists almost surely of a pure-point component and an absolutely continuous component. The former is the diffraction measure of the expectation E[omega], while the inverse Fourier transform of the absolutely continuous component of omega turns 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 of Lambda Then we provide a sufficient condition that the diffraction measure of a random field on a model set is still pure-point.
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