We study the statistical mechanics of the log-gas, or β-ensemble, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on the next order energy that are valid down to microscopic length scales. To our knowledge, this is the first time that this kind of a local quantity has been controlled for the log-gas. Simultaneously, we exhibit a control on fluctuations of linear statistics that is valid at all mesoscales using Johansson's method and a transport approach. Using these local laws, we are able to exhibit for the first time a CLT at arbitrary mesoscales, improving upon previous results that were true only for power mesoscales.

中文翻译：

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β-系综的局部定律和介观 CLT

我们研究一般势能和反温度的对数气体或 β 系综的统计力学。通过自举程序，我们证明了下一阶能量的局部定律在微观长度尺度上是有效的。据我们所知，这是首次对原木气体进行这种局部量控制。同时，我们使用 Johansson 方法和传输方法展示了对线性统计波动的控制，该控制在所有介尺度上都有效。利用这些局部定律，我们能够首次展示任意介尺度的 CLT，改进了之前仅适用于功率介尺度的结果。