We investigate Sine$_\beta$, the universal point process arising as the thermodynamic limit of the microscopic scale behavior in the bulk of one-dimensional log-gases, or $\beta$-ensembles, at inverse temperature $\beta>0$. We adopt a statistical physics perspective, and give a description of Sine$_\beta$ using the Dobrushin-Lanford-Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sine$_\beta$ to a compact set, conditionally to the exterior configuration, reads as a Gibbs measure given by a finite log-gas in a potential generated by the exterior configuration. Moreover, we show that Sine$_\beta$ is number-rigid and tolerant in the sense of Ghosh-Peres, i.e. the number, but not the position, of particles lying inside a compact set is a deterministic function of the exterior configuration. Our proof of the rigidity differs from the usual strategy and is robust enough to include more general long range interactions in arbitrary dimension.

中文翻译：

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Sine-Beta 过程的 DLR 方程和刚性

我们研究了正弦$_\beta$，作为在逆温度$\beta>0 下大量一维对数气体或$\beta$-集合中微观尺度行为的热力学极限而产生的通用点过程$. 我们采用统计物理学的观点，并通过证明它满足 DLR 方程来使用 Dobrushin-Lanford-Ruelle (DLR) 形式主义给出 Sine$_\beta$ 的描述：Sine$_\beta$ 对紧凑的限制有条件地设置为外部配置，读取为由外部配置产生的势中的有限对数气体给出的吉布斯测度。此外，我们表明 Sine$_\beta$ 在 Ghosh-Peres 的意义上是数字刚性和宽容的，即位于紧集内的粒子的数量而不是位置是外部配置的确定性函数。