Nodal deficiency of random spherical harmonics in presence of boundary

Valentina Cammarota, D. Marinucci, I. Wigman
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引用次数: 2

Abstract

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.
存在边界时随机球谐波的节点缺陷
考虑沿赤道满足狄利克雷边界条件的半球上拉普拉斯特征函数的随机高斯模型。对于这个模型,我们发现了相应的零密度函数在短距离(边界附近)和远距离(远离边界)状态下的精确渐近律。作为推论,我们能够找到相对于随机球谐波的旋转不变模型,该集合的总节点长度的对数负偏差。Jean Bourgain的研究,以及他对拉普拉斯特征函数的节点几何的热情方法,在该领域和当前的趋势中产生了至关重要的影响。他在谱相关性(Krishnapur, Kurlberg和Wigman(2013)中的定理2.2)方面的工作,以及与Bombieri (Bourgain和Bombieri(2015))的合作,为在算术风格的表面(如环面或方形)上定义的函数的节点长度的积极研究打开了一道门。此外,Bourgain关于整体拉普拉斯特征函数的研究(Bourgain(2014))也吸引了光谱相关性,允许从随机高斯对应物中推断确定性结果。
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