全维分级库仑气体的基态和超均匀性
Ground States And Hyperuniformity Of The Hierarchical Coulomb Gas In All Dimensions
摘要：
具有库仑相互作用的随机点过程出现在统计力学、随机矩阵和最优化问题的各种自然例子中。通常，由于它们的自然排斥，这样的系统表现出显着的超均匀性，也就是说，在任何给定区域中着陆的点的数量与一组i.i.d.相比以小得多的尺度波动。随机点在Jancovici、Lebowitz、Manificat、Martin和Yalcin（'80，' 83，'93）的著作中出现的一个著名的物理学猜想指出，落在集合中的点的数量的方差应该像表面积而不是体积一样增长，而不是i.i.d.。随机点在最近的一项美丽的工作中，Chatterjee（2017）首次证明了库仑型系统的三维结果，称为分层库仑气体，灵感来自戴森的伊辛铁磁体分层模型。然而，尺寸大于3的案件仍然悬而未决。在本文中，我们建立了正确的波动行为的对数因子在所有维度大于三，分层模型。使用类似的方法，我们也证明了尖锐的方差界光滑的线性统计量，这是未知的任何尺寸大于2。一个关键的中间步骤是获得精确的结果，这些模型的行为可以被解释为层次类似物的各种晶体结构预测的能量最小化系统的基态，并可能是独立的利益。
英文摘要：
Stochastic point processes with Coulomb interactions arise in various natural examples of statistical mechanics, random matrices and optimization problems. Often such systems due to their natural repulsion exhibit remarkable hyperuniformity properties, that is, the number of points landing in any given region fluctuates at a much smaller scale compared to that of a set of i.i.d. random points. A well known conjecture from physics appearing in the works of Jancovici, Lebowitz, Manificat, Martin, and Yalcin ('80,'83,'93), states that the variance of the number of points landing in a set should grow like the surface area instead of the volume unlike i.i.d. random points. In a recent beautiful work, Chatterjee (2017) gave the first proof of such a result in dimension three for a Coulomb type system, known as the hierarchical Coulomb gas, inspired by Dyson's hierarchical model of the Ising ferromagnet. However the case of dimensions greater than three had remained open. In this paper, we establish the correct fluctuation behavior up to logarithmic factors in all dimensions greater than three, for the hierarchical model. Using similar methods, we also prove sharp variance bounds for smooth linear statistics which were unknown in any dimension bigger than two. A key intermediate step is to obtain precise results about the ground states of such models whose behavior can be interpreted as hierarchical analogues of various crystalline conjectures predicted for energy minimizing systems, and could be of independent interest.
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