大亏格随机曲面上本征函数的离域化
Delocalisation of eigenfunctions on large genus random surfaces
摘要：
我们证明了在一个紧凑的双曲曲面上的拉普拉斯算子的特征函数离域的几何参数依赖于表面上的短闭测地线的数量。特别是，我们表明，一个$L^2$规范化的本征函数限制到一个可测量的子集的表面有平方$L^2$ -规范$\varepsilon>0$，只有当一组有一个相对较大的尺寸-指数的几何参数。对于随机表面的威尔-彼得森概率测度，然后，我们表明，与高概率$g\to\infty$，该集的大小必须至少是表面的亏格依赖于本征值和$\varepsilon$的一些权力。
英文摘要：
We prove that eigenfunctions of the Laplacian on a compact hyperbolic surface delocalise in terms of a geometric parameter dependent upon the number of short closed geodesics on the surface. In particular, we show that an $L^2$ normalised eigenfunction restricted to a measurable subset of the surface has squared $L^2$-norm $\varepsilon>0$, only if the set has a relatively large size -- exponential in the geometric parameter. For random surfaces with respect to the Weil-Petersson probability measure, we then show, with high probability as $g\to\infty$, that the size of the set must be at least the genus of the surface to some power dependent upon the eigenvalue and $\varepsilon$.
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