论文标题
在边界存在下随机球形谐波的淋巴结缺乏
Nodal deficiency of random spherical harmonics in presence of boundary
论文作者
论文摘要
我们考虑了半球上拉普拉斯本征函数的随机高斯模型，该模型满足了沿赤道的差异边界条件。对于此模型，我们发现在短范围（边界）和远距离（远离边界）方案的相应零密度函数的精确渐近定律。作为推论，我们能够在相对与随机球形谐波的旋转模型相对旋转模型中找到该集合的总淋巴结长度的对数负偏差。 让·布尔加因（Jean Bourgain）的研究以及他对拉普拉斯特征函数节点几何形状的热情方法，对该领域和当前趋势产生了至关重要的影响。他在光谱相关性方面的作品（Krishnapur，Kurlberg和Wigman（2013）的定理2.2）和与Bombieri（Bourgain and Bombieri（2015））的作品为正在进行的持续研究开辟了一扇门，以研究对在算术表面上定义的功能的持续研究，例如Torus或Square或Square或Square。此外，布尔加因在托拉拉（Toral Laplace）特征函数上的工作（Bourgain（2014））也吸引了光谱相关性，可以从随机高斯同行中推断出确定性的结果。
We consider a random Gaussian model of Laplace eigenfunctions on the hemisphere satisfying the Dirichlet boundary conditions along the equator. For this model we find a precise asymptotic law for the corresponding zero density functions, in both short range (around the boundary) and long range (far away from the boundary) regimes. As a corollary, we were able to find a logarithmic negative bias for the total nodal length of this ensemble relatively to the rotation invariant model of random spherical harmonics. Jean Bourgain's research, and his enthusiastic approach to the nodal geometry of Laplace eigenfunctions, has made a crucial impact in the field and the current trends within. His works on the spectral correlations (Theorem 2.2 in Krishnapur, Kurlberg and Wigman (2013)) and joint with Bombieri (Bourgain and Bombieri (2015)) have opened a door for an active ongoing research on the nodal length of functions defined on surfaces of arithmetic flavour, like the torus or the square. Further, Bourgain's work on toral Laplace eigenfunctions (Bourgain (2014)), also appealing to spectral correlations, allowed for inferring deterministic results from their random Gaussian counterparts.
