发表机构
The University of Texas at Austin; University of Edinburgh; Carnegie Mellon University（德克萨斯大学奥斯汀分校; 爱丁堡大学; 卡内基梅隆大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对希尔伯特-施密特区域的扩散对数/里斯气体，通过相关分析得到严格平均场估计，扩展了对数配分函数估计范围并建立了相关中心极限定理等结果。
AI 中文摘要
我们研究固定温度下的对数气体和里斯气体，在减去相对于规定背景律的主导平均场贡献之后的相关性质。对于希尔伯特-施密特区域中的排斥相互作用，我们证明了N-一致界，以及由此产生的调制配分函数收敛到由中心化相互作用算子的Carleman-Fredholm行列式表示的归一化量的定量结果。我们表明希尔伯特-施密特阈值是严格的，并获得了在该阈值及以上发散速率的显式下界；这些速率预计是非最优的。证明结合了正定截断、低/高频分解，以及典型二阶U-统计量的指数不等式和高斯混沌渐近分析。作为推论，我们在调制自由能方法中建立了具有严格O(N⁻¹)加性尺度的熵对易子估计，得到了静态联合线性统计量中心极限定理，以及有限个时刻联合线性统计量的动态中心极限定理。这将前两位作者的对数配分函数估计扩展到了完整的里斯希尔伯特-施密特范围，并确定了极限行列式归一化量。对于足够小的逆温度下的吸引对数相互作用，我们也证明了类似的结果。
英文摘要
We study fixed-temperature logarithmic and Riesz gases after subtracting the leading mean-field contribution relative to a prescribed background law. For repulsive interactions in the Hilbert--Schmidt regime, we prove $N$-uniform bounds and quantitative convergence of the resulting modulated partition function to a normalization expressed by the Carleman--Fredholm determinant of the centered interaction operator. We show that the Hilbert--Schmidt threshold is sharp and obtain explicit lower bounds on the rate of divergence at and above it; these rates are expected to be nonoptimal. The proof combines positive-definite truncations and a low/high-frequency decomposition with exponential inequalities and Gaussian-chaos asymptotics for canonical degree-two $U$-statistics. As consequences, we establish entropic commutator estimates with the sharp $O(N^{-1})$ additive scale in the modulated-free-energy method, a static joint linear-statistics central limit theorem, and a dynamical central limit theorem for joint linear statistics at finitely many times. This extends the logarithmic partition-function estimates of the first two authors to the full Riesz Hilbert--Schmidt range and identifies the limiting determinant normalization. For the attractive logarithmic interaction at sufficiently small inverse temperature, we also prove analogous results.
Comments68 pages
