论文标题
库仑气体的拥挤和分离估计值
Overcrowding and Separation Estimates for the Coulomb Gas
论文作者
论文摘要
我们证明了库仑气体在任何维度$ d \ geq 2 $中的几个结果，这些结果来自各向同性平均，这是一种基于牛顿定理的传输方法。首先，我们证明了一项高密度的Jancovici-Lebowitz-Manifitz法律，扩大了Armstrong和Serfaty的微观密度界限，并建立了严格的次高帽子尾巴，以付费超过尺寸$ 2 $。在液滴的边缘证明了微观限制点过程的存在。接下来，我们证明了合并点的$ K $ - 点相关功能的最佳上限，包括$ k = 1 $的库仑气体的韦格纳估计。我们证明了正确重新缩放的$ k $ th最小粒子间隙的紧密度，并在$ d = 2 $中确定了正确的订单，并在$ d \ geq 3 $中确定了三个任期的扩展，以及上下尾巴的估计。特别是，我们将Ameur和Romero确定的二维“完美冻结制度”扩展到更高的维度。最后，我们给出了积极的电荷差异界限，这是液滴边界附近的最先进的状态，并证明了Laughlin状态在分数量子厅效应中的不可压缩性，从大型显微镜尺度开始。利用刚性的线性统计波动，我们展示了如何升级正差界限到对某些区域中绝对差异的估计。
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.
