标题： 围绕库仑气体热平衡测度的集中不等式 显示英文标题

标题： Concentration inequality around the thermal equilibrium measure of Coulomb gases

摘要： 本文讨论在中间温度范围内 的库仑气体，其中在微观层面上没有观察到结构，但质量被限制在一个紧集内。 我们的主要结果是一个围绕热平衡测度的集中不等式，表明以指数接近$1,$的概率，经验测度与热平衡测度$\mathcal{O}\left( \frac{1}{N^{\frac{1}{d}}}\right)$接近。 我们还证明了这个集中不等式在某种意义上是最佳的。 主要的新工具是函数不等式，它们允许我们在某些测度不具有紧支撑的情况下，将测度的有界利普希茨范数与其$H^{-1}$范数进行比较。 显示英文摘要

摘要： 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.