论文标题
库仑和里斯气体：已知和未知的
Coulomb and Riesz gases: The known and the unknown
论文作者
论文摘要
我们回顾有关库仑和里斯气体的数学特性的已知，未知和期望。这些描述了$ \ mathbb {r}^d $与riesz潜在$ \ pm | x | x |^{ - s} $中的点的无限配置我们的演讲遵循统计力学的标准观点，但我们还提到了这些系统如何在其他重要情况下出现（例如，在随机矩阵理论中）。本文中解决的主要问题是如何正确定义相关的无限点过程并使用一些（重归于）平衡方程来表征它。这在远程$ s <d $中在很大程度上开放。为了方便读者，我们提供了简短案例$ s> d $中已知内容的详细信息。在最后一部分中，我们讨论了相变并提及基于物理理由的期望。
We review what is known, unknown and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in $\mathbb{R}^d$ interacting with the Riesz potential $\pm |x|^{-s}$ (resp. $-\log|x|$ for $s=0$). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g. in random matrix theory). The main question addressed in the article is how to properly define the associated infinite point process and characterize it using some (renormalized) equilibrium equation. This is largely open in the long range case $s<d$. For the convenience of the reader we give the detail of what is known in the short range case $s>d$. In the last part we discuss phase transitions and mention what is expected on physical grounds.
