For a compact hyperbolic surface, we define a smooth linear statistic, mimicking the number of Laplace eigenvalues in a short energy window. We study the variance of this statistic, when averaged over the moduli space \(\mathcal{M}_{g}\) of all genus g surfaces with respect to the Weil-Petersson measure. We show that in the double limit, first taking the large genus limit and then the short window limit, we recover GOE statistics for the variance. The proof makes essential use of Mirzakhani’s integration formula.

中文翻译：

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大亏格曲面模空间的GOE统计

对于紧凑双曲曲面，我们定义了一个平滑的线性统计量，模仿短能量窗口中拉普拉斯特征值的数量。我们研究了该统计量的方差，当相对于 Weil-Petersson 测度在所有属g曲面的模空间\(\mathcal{M}_{g}\)上进行平均时。我们证明，在双重极限中，首先采用大属极限，然后采用短窗口极限，我们恢复了方差的 GOE 统计量。该证明充分利用了 Mirzakhani 的积分公式。