紧曲面的随机覆盖与光滑线性谱统计量
Random covers of compact surfaces and smooth linear spectral statistics
摘要：
考虑任意紧双曲曲面$X$的随机n-覆盖$X_n$.我们表明，在大n制度和小窗口限制，由酉阿贝尔字符扭曲的拉普拉斯光滑谱统计的方差，服从GOE和GUE随机矩阵的普遍规律，取决于字符保持或破坏时间反演对称性。我们还证明了紧线性群中高维扭值的一个推广。这些结果证实了Berry的一个猜想，并且是Rudnick最近关于随机表面的Weil-Petersson模型的工作的离散模拟。
英文摘要：
We consider random n-covers $X_n$ of an arbitrary compact hyperbolic surface $X$. We show that in the large n regime and small window limit, the variance of the smooth spectral statistics of the Laplacian twisted by a unitary abelian character, obey the universal laws of GOE and GUE random matrices, depending on wether the character preserves or breaks the time reversal symmetry. We also prove a generalization for higher dimensional twists valued in compact linear groups. These results confirm a conjecture of Berry and is a discrete analog of a recent work of Rudnick for the Weil-Petersson model of random surfaces.
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