论文标题
在任何温度下的高斯波动和库仑气体的自由能量膨胀
Gaussian Fluctuations and Free Energy Expansion for Coulomb Gases at Any Temperature
论文作者
论文摘要
我们获得了在任何维度和广泛的温度状态下的库仑气体波动的浓度估计值，包括可能取决于点数的非常小且非常大的温度状态。我们获得了一个完整的中央限制定理，以在维度2中线性统计的波动，首次有效到显微镜，并且可能很小或非常大的温度。 我们表明，也可以在任何较大的尺寸中获得类似的CLT在“无相转换”假设上的任何较大尺寸中获得，只要人们可以获得足够精确的误差率来扩展自由能 - 在任何维度上都可以获得扩展，但是所获得的速率是如此之遥得多。 这些CLT可以解释为与高斯自由场的收敛。 一旦测试功能的生命比我们以前的工作\ cite {as}引入的温度相关的最小量表$ρ_β$，所有结果就会有效。
We obtain concentration estimates for the fluctuations of Coulomb gases in any dimension and in a broad temperature regime, including very small and very large temperature regimes which may depend on the number of points. We obtain a full Central Limit Theorem for the fluctuations of linear statistics in dimension 2, valid for the first time down to microscales and for possibly very small or very large temperatures. We show that a similar CLT can also be obtained in any larger dimension conditional on a "no phase-transition" assumption, as soon as one can obtain a precise enough error rate for the expansion of the free energy -- an expansion is obtained in any dimension, but the obtained rate is so far not good enough to conclude. These CLTs can be interpreted as a convergence to the Gaussian Free Field. All the results are valid as soon as the test-function lives on a larger scale than the temperature-dependent minimal scale $ρ_β$ introduced in our previous work \cite{as}.
