Gaussian fluctuations and free energy expansion for Coulomb gases at any temperature
Gaussian fluctuations and free energy expansion for Coulomb gases at any temperature
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DOI：
10.1214/22-aihp1285
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发表时间：
2020-03
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影响因子：
--
通讯作者：
S. Serfaty
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作者：
S. Serfaty
我们得到浓度估计的波动库仑气体在任何尺寸和在一个广泛的温度制度，包括非常小和非常大的温度制度，这可能取决于点的数量。我们得到一个完整的中心极限定理的波动的线性统计在2维，有效的第一次下降到微尺度和可能非常小或非常大的温度。我们表明，一个类似的CLT也可以在任何更大的尺寸条件下获得一个“无相变”的假设，只要一个人可以获得足够精确的误差率的自由能的扩展-扩展是在任何尺寸，但获得的速度是迄今为止还不够好的结论。这些CLT可以被解释为高斯自由场的收敛。所有的结果是有效的，只要测试功能的生活在一个更大的规模比温度依赖的最小规模$\rho_\beta $在我们以前的工作\cite {as}。
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 $\rho_\beta$ introduced in our previous work \cite{as}.