For the two-dimensional one-component Coulomb plasma, we derive an asymptotic expansion of the free energy up to order $N$, the number of particles of the gas, with an effective error bound $N^{1-\kappa}$ for some constant $\kappa > 0$. This expansion is based on approximating the Coulomb gas by a quasi-free Yukawa gas. Further, we prove that the fluctuations of the linear statistics are given by a Gaussian free field at any positive temperature. Our proof of this central limit theorem uses a loop equation for the Coulomb gas, the free energy asymptotics, and rigidity bounds on the local density fluctuations of the Coulomb gas, which we obtained in a previous paper.

中文翻译：

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二维库仑等离子体：准自由近似和中心极限定理

对于二维单分量库仑等离子体，我们推导出自由能的渐近膨胀至 $N$ 阶，即气体粒子的数量，有效误差界为 $N^{1-\kappa}$对于一些常数 $\kappa > 0$。这种扩展基于通过准自由汤川气体近似库仑气体。此外，我们证明了线性统计的波动是由任何正温度下的高斯自由场给出的。我们对这个中心极限定理的证明使用了库仑气体的循环方程、自由能渐近性和库仑气体局部密度波动的刚性边界，这是我们在之前的一篇论文中获得的。