SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 181-220, January 2021.
We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a large deviations technique known as the Gibbs conditioning principle. The technical part amounts to establishing that the conditioning ensemble is an $I$-continuity set of the energy. This leads to characterizing the conditioned equilibrium measure as the solution of a modified variational problem. For simplicity, we focus on linear statistics and on quadratic statistics constraints. Finally, we numerically illustrate our predictions and explore cases in which no explicit solution is known. For this, we use a generalized hybrid Monte Carlo algorithm for sampling from the conditioned distribution for a finite but large system.

中文翻译：

---

约束下的库仑气体：一些理论和数值结果

SIAM数学分析杂志，第53卷，第1期，第181-220页，2021年1月。
我们考虑库仑气体模型，当颗粒数量变大时，经验度量通常将其集中在平衡度量上，以使静电能量最小。我们研究了气体以罕见事件为条件时的行为。我们首先表明，由于显着的因式分解，二次约束和线性约束的特殊情况是完全可以解决的，然后这种条件具有使粒子云移动而不变形的简单效果。为了解决更一般的情况，我们依靠称为Gibbs条件原理的大偏差技术执行理论渐近分析。技术部分相当于确定调节集合是能量的$ I $连续性集合。这导致将条件均衡度量表征为修正的变分问题的解决方案。为简单起见，我们关注线性统计和二次统计约束。最后，我们以数字方式说明我们的预测，并探索其中尚无明确解决方案的情况。为此，我们使用广义混合蒙特卡罗算法从有限但较大的系统的条件分布中进行采样。