论文标题
库仑气体的热平衡度量周围的浓度不平等
Concentration inequality around the thermal equilibrium measure of Coulomb gases
论文作者
论文摘要
本文涉及中等温度状态下的库仑气体，其中在微观水平上未观察到结构，但质量限制在紧凑的集合中。我们的主要结果是围绕热平衡度量的浓度不平等，指出，概率接近$ 1，$ $ $ \ nathcal是$ \ MATHCAL {O} \ left（\ frac {1} {n^{n^{\ frac {\ frac {1}} {d}} {d}}} \ right）我们还证明这种集中不平等在某种意义上是最佳的。主要的新工具是功能上的不平等，使我们能够将有限的Lipschitz标准与其$ H^{ - 1} $ NORM进行比较，在某些情况下，该度量没有紧凑的支持。
This article deals with Coulomb gases at an intermediate temperature regime, in which no structure is observed at the microscopic level, but the mass in confined to a compact set. Our main result is a concentration inequality around the thermal equilibrium measure, stating that with probability exponentially close to $1,$ the empirical measure is $\mathcal{O}\left( \frac{1}{N^{\frac{1}{d}}}\right)$ close to the thermal equilibrium measure. We also prove that this concentration inequality is optimal in some sense. The main new tool are functional inequalities that allow us to compare the bounded Lipschitz norm of a measure to its $H^{-1}$ norm in some cases when the measure does not have compact support.
