库仑气体的过拥挤和分离估计
Overcrowding and Separation Estimates for the Coulomb Gas
摘要：
我们证明了几个结果的库仑气体在任何尺寸$d \geq 2$，遵循各向同性平均，基于牛顿定理的运输方法。首先，我们证明了一个高密度的Jancovici-Lebowitz-Manificat定律，扩展了Armstrong和Serfaty的微观密度边界，并建立了$2$维电荷过剩的严格亚高斯尾。证明了液滴边缘存在微观极限点过程。接下来，我们证明了最佳上界的$k$点关联函数的合并点，包括韦格纳估计的库仑气体$k=1$。我们证明了适当重新缩放的$k$ th最小粒子间隙的紧密性，确定了$d=2$中的正确顺序和$d \geq 3$中的三项扩展，以及上下尾估计。特别是，我们扩展了二维的“完美冻结制度”确定的Ameur和罗梅罗到更高的维度。最后，我们给出了液滴边界附近的正电荷差异边界，并证明了分数量子霍尔效应中Laughlin态的不可压缩性。利用光滑线性统计量波动的刚性，我们展示了如何将正偏差界升级为对某些区域绝对偏差的估计。
英文摘要：
We prove several results for the Coulomb gas in any dimension $d \geq 2$ that follow from isotropic averaging, a transport method based on Newton's theorem. First, we prove a high-density Jancovici-Lebowitz-Manificat law, extending the microscopic density bounds of Armstrong and Serfaty and establishing strictly sub-Gaussian tails for charge excess in dimension $2$. The existence of microscopic limiting point processes is proved at the edge of the droplet. Next, we prove optimal upper bounds on the $k$-point correlation function for merging points, including a Wegner estimate for the Coulomb gas for $k=1$. We prove the tightness of the properly rescaled $k$th minimal particle gap, identifying the correct order in $d=2$ and a three term expansion in $d \geq 3$, as well as upper and lower tail estimates. In particular, we extend the two-dimensional "perfect-freezing regime" identified by Ameur and Romero to higher dimensions. Finally, we give positive charge discrepancy bounds which are state of the art near the droplet boundary and prove incompressibility of Laughlin states in the fractional quantum Hall effect, starting at large microscopic scales. Using rigidity for fluctuations of smooth linear statistics, we show how to upgrade positive discrepancy bounds to estimates on the absolute discrepancy in certain regions.
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