We prove several results for the Coulomb gas in any dimension $d\ge 2$ 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 kth minimal particle gap, identifying the correct order in $d=2$ and a three term expansion in $d\ge 3$, 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.

中文翻译：

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库仑气体的过度拥挤和分离估计

我们证明了库仑气体在任何维度上的几个结果$d\ge 2$遵循各向同性平均，这是一种基于牛顿定理的传输方法。首先，我们证明了高密度 Jancovici-Lebowitz-Manificat 定律，扩展了 Armstrong 和 Serfaty 的微观密度界限，并为 2 维电荷过剩建立了严格的亚高斯尾部。在边缘证明了微观极限点过程的存在的液滴。接下来，我们证明合并点的k点相关函数的最佳上限，包括库仑气体的韦格纳估计$k=1$。我们证明了正确重新调整的第 k个最小粒子间隙的紧密性，确定了正确的顺序$d=2$以及三期扩张$d\ge 3$，以及上尾和下尾估计。特别是，我们将 Ameur 和 Romero 确定的二维“完美冷冻机制”扩展到更高的维度。最后，我们给出了液滴边界附近最先进的正电荷差异边界，并从大微观尺度开始证明了分数量子霍尔效应中劳克林态的不可压缩性。使用平滑线性统计波动的刚性，我们展示了如何将正差异界限升级为对某些区域的绝对差异的估计。