We study the fluctuations of smooth linear statistics of Laplace eigenvalues of compact hyperbolic surfaces lying in short energy windows, when averaged over the moduli space of surfaces of a given genus. The average is taken with respect to the Weil–Petersson measure. We show that first taking the large genus limit, then a short window limit, the distribution tends to a Gaussian. The variance was recently shown to be given by the corresponding quantity for the Gaussian Orthogonal Ensemble (GOE), and the Gaussian fluctuations are also consistent with those in Random Matrix Theory, as conjectured in the physics literature for a fixed surface.

中文翻译：

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大亏格随机面上线性特征值统计的中心极限定理

我们研究了位于短能量窗口中的紧致双曲曲面拉普拉斯特征值的平滑线性统计的波动，当在给定曲面的模空间上进行平均时。平均值是根据 Weil-Petersson 测度得出的。我们证明，首先采用大属限制，然后采用短窗口限制，分布趋于高斯分布。最近表明，方差是由高斯正交系综 (GOE) 的相应量给出的，并且高斯波动也与随机矩阵理论中的波动一致，正如物理文献中对固定表面的推测。