超均匀Delone实现与刚性
Hyperuniform Delone Realizations and Rigidity
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中文总结 AI 辅助
该研究证明超均匀Delone点过程的可测实现定理，在不改变可测动力学下实现任意高阶低频抑制，还兼具多种刚性性质，并推广到欧几里得运动作用与一维流情形。
中文摘要 AI 辅助
我们证明了超均匀Delone点过程的可测实现定理。在维度d≥2时，对于每个指定的q≥1，ℝᵈ的每个本质自由遍历保概率测度（p.m.p.）作用，在每个足够大的指定强度下，都存在一个生成性Delone实现，其返回时间点过程η与原始作用可测同构，且其Bartlett谱σ_η满足σ_η(B_ε)=o(ε^{2q})（ε↓0）。因此可以在不改变指定可测动力学的前提下，施加任意高阶的有限阶低频抑制。这些实现还可被选择为具有面阶球方差和对任意指定有限阶的线性刚性，同时具备最大刚性，且几乎必然与格点有界位移等价。对于平移子作用为遍历的本质自由欧几里得运动作用，构造可以是各向同性的且V-遍历的，因此也是V-弱混合的。在一维情形下，每个本质自由遍历流都存在生成性Delone实现，具有对数区间偏差、最大刚性，且Bartlett谱在原点处近二次衰减。
英文摘要
We prove a measurable realization theorem for hyperuniform Delone point processes. In dimensions \(d\geq2\), for every prescribed \(q\geq1\), every essentially free ergodic p.m.p.\ action of \(\mathbb R^d\) admits, at every sufficiently large prescribed intensity, a generating Delone realization whose return-time point process \(η\) is measurably isomorphic to the original action and whose Bartlett spectrum \(σ_η\) satisfies \[ σ_η(B_\varepsilon)=o(\varepsilon^{2q}) \qquad(\varepsilon\downarrow0). \] Thus arbitrarily high finite-order low-frequency suppression can be imposed without changing the prescribed measurable dynamics. The same realizations can be chosen with surface-order ball variance and linear rigidity to any prescribed finite order, while also being maximally rigid and almost surely bounded-displacement equivalent to a lattice. For essentially free Euclidean-motion actions whose translation subaction is ergodic, the construction can be made isotropic and \(V\)-ergodic, and hence \(V\)-weakly mixing. In dimension one, every essentially free ergodic flow admits generating Delone realizations with logarithmic interval discrepancy, maximal rigidity, and near-quadratic decay of the Bartlett spectrum at the origin.
