三维分层库仑气体的刚性
Rigidity of the three-dimensional hierarchical Coulomb gas
摘要：
在欧几里得空间中的一个随机点集叫做“刚性”或“超均匀”，如果落在任何给定区域内的点的数目的波动明显小于一个独立同分布的点集的相应数目。随机点近年来，由于它在随机矩阵理论、库仑气体理论和随机解析函数的零点中的出现，这一现象受到了相当大的关注。然而，大多数已发表的结果都是一维和二维的。本文首次证明了三维库仑型系统的超均匀性，称为等级库仑气体。这是实际三维库仑气体的简化版本。该模型中的相互作用势受戴森的伊辛铁磁体分层模型的启发，具有分层结构，并且局部近似库仑势。超均匀性被证明在宏观和微观尺度上，与上界和下界的波动，匹配的对数因子的顺序。波动有立方根行为，与一个众所周知的预测三维库仑气体。为了完整性，类似的结果也证明了二维分层库仑气体和一维分层日志气体。
英文摘要：
A random set of points in Euclidean space is called `rigid' or `hyperuniform' if the number of points falling inside any given region has significantly smaller fluctuations than the corresponding number for a set of i.i.d. random points. This phenomenon has received considerable attention in recent years, due to its appearance in random matrix theory, the theory of Coulomb gases and zeros of random analytic functions. However, most of the published results are in dimensions one and two. This paper gives the first proof of hyperuniformity in a Coulomb type system in dimension three, known as the hierarchical Coulomb gas. This is a simplified version of the actual 3D Coulomb gas. The interaction potential in this model, inspired by Dyson's hierarchical model of the Ising ferromagnet, has a hierarchical structure and is locally an approximation of the Coulomb potential. Hyperuniformity is proved at both macroscopic and microscopic scales, with upper and lower bounds for the order of fluctuations that match up to logarithmic factors. The fluctuations have cube-root behavior, in agreement with a well-known prediction for the 3D Coulomb gas. For completeness, analogous results are also proved for the 2D hierarchical Coulomb gas and the 1D hierarchical log gas.
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