论文标题
在大属随机表面上特征函数的定位
Delocalisation of eigenfunctions on large genus random surfaces
论文作者
论文摘要
我们证明，laplacian在紧凑的双曲表面脱位上的本征函数取决于几何参数取决于表面上短闭合的大地测量学的数量。特别是，我们表明$ l^2 $归一化的特征函数仅限于表面的可测量子集已平方$ l^2 $ -NORM $ \ VAREPSILON> 0 $，仅当该集合具有相对较大的尺寸 - 几何参数中的指数。对于随机表面相对于Weil-Petersson的概率度量，我们以高概率为$ g \ to \ infty $，表明该集合的大小至少必须是表面属，这些功率依赖于特征值和$ \ \ \ Varepsilon $。
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 $\varepsilon$.
