用库仑气体设计随机积分算法，无需精确调参仍可高效逼近积分结果。
Quenched large deviations for Monte Carlo integration with Coulomb gases
- 利用库仑气体粒子间斥力机制构造随机积分点集。
- 即使使用廉价蒙特卡洛预处理近似势能，仍保持快速大偏差率。
- 适用于需高精度积分但难以调参的复杂分布场景。
吉布斯测度（如库仑气体）常用于建模相互作用粒子系统。近期研究提出将吉布斯测度作为相对于目标测度π在ℝᵈ上的随机数值积分算法，其直观依据是粒子间的排斥性有助于降低积分误差。该方法的主要挑战在于调节相互作用核与约束势能，使系统的平衡测度恰好为π。通常需通过另一蒙特卡洛近似来计算势能（即相互作用核对π的积分）。本文基于Garcia–Zelada（2019）的大偏差方法，证明对势能的随机近似仍能保持快速大偏差原理，从而确保该积分算法优于独立或马尔可夫型求积法。对于非奇异相互作用核，仅需最小假设，近似可来自计算成本低廉的蒙特卡洛预处理；对于库仑相互作用核，要求近似基于另一个吉布斯测度，并在过程中建立了势能近似一致收敛的控制边界。
原文摘要 · Abstract (English)
Gibbs measures, such as Coulomb gases, are popular in modelling systems of interacting particles. Recently, we proposed to use Gibbs measures as randomized numerical integration algorithms with respect to a target measure $π$ on $\mathbb R^d$, following the heuristics that repulsiveness between particles should help reduce integration errors. A major issue in this approach is to tune the interaction kernel and confining potential of the Gibbs measure, so that the equilibrium measure of the system is the target distribution $π$. Doing so usually requires another Monte Carlo approximation of the \emph{potential}, i.e. the integral of the interaction kernel with respect to $π$. Using the methodology of large deviations from Garcia--Zelada (2019), we show that a random approximation of the potential preserves the fast large deviation principle that guarantees the proposed integration algorithm to outperform independent or Markov quadratures. For non-singular interaction kernels, we make minimal assumptions on this random approximation, which can be the result of a computationally cheap Monte Carlo preprocessing. For the Coulomb interaction kernel, we need the approximation to be based on another Gibbs measure, and we prove in passing a control on the uniform convergence of the approximation of the potential.
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