具有奇异相互作用的非均匀气体：广义圆律和非均匀重整化能
The heterogeneous gas with singular interaction: Generalized circular law and heterogeneous renormalized energy
摘要：
介绍并分析了具有随机电荷分布和一般外约束势的$d$维库仑气体。我们证明了这些气体满足大偏差原理。对速率函数（能量的主导项）的最小值的分析表明，在平衡时，粒子分布是广义的圆形定律（即具有球形支撑但不一定均匀分布）。在经典静电外势中，存在无穷多个速率函数的极小元。最可能的宏观构型是无序分布，其中粒子均匀分布（对于$d=2$，圆形定律），并且电荷与粒子的位置无关。一般的电荷相关的限制势展开这种简并的情况：相反，粒子密度是不均匀的，粒子自发地组织根据他们的电荷。在这幅图中，经典静电势表现为一个跃迁，在这个跃迁处，有序性消失了。子领导项的能量推导：我们表明，这些都与运营商，推广库仑重整化能量，其中包括异质性的收费。这种异质重整化能量告诉我们粒子的微观排列，这些排列是非标准的，强烈依赖于电荷，并且包括渐进和不规则的晶格。
英文摘要：
We introduce and analyze $d$ dimensional Coulomb gases with random charge distribution and general external confining potential. We show that these gases satisfy a large deviations principle. The analysis of the minima of the rate function (which is the leading term of the energy) reveals that at equilibrium, the particle distribution is a generalized circular law (i.e. with spherical support but non-necessarily uniform distribution). In the classical electrostatic external potential, there are infinitely many minimizers of the rate function. The most likely macroscopic configuration is a disordered distribution in which particles are uniformly distributed (for $d=2$, the circular law), and charges are independent of the positions of the particles. General charge-dependent confining potentials unfold this degenerate situation: in contrast, the particle density is not uniform, and particles spontaneously organize according to their charge. In that picture the classical electrostatic potential appears as a transition at which order is lost. Sub-leading terms of the energy are derived: we show that these are related to an operator, generalizing the Coulomb renormalized energy, which incorporates the heterogeneous nature of the charges. This heterogeneous renormalized energy informs us about the microscopic arrangements of the particles, which are non-standard, strongly depending on the charges, and include progressive and irregular lattices.
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