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 relative 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 the work of Krishnapur et al. [Ann. Math. 177(2), 699–737 (2013)]} and Bombieri and Bourgain [Int. Math. Res. Not. (IMRN) 11, 3343–3407 (2015)] have opened a door for an active ongoing research on the nodal length of functions defined on surfaces of arithmetic flavor, such as the torus or the square. Furthermore, Bourgain’s work [J. Bourgain, Isr. J. Math. 201(2), 611–630 (2014)] on toral Laplace eigenfunctions, also appealing to spectral correlations, allowed for inferring deterministic results from their random Gaussian counterparts.

中文翻译：

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存在边界时随机球谐函数的节点缺乏

我们考虑半球上的Laplace特征函数的随机高斯模型，该模型满足赤道沿Dirichlet的边界条件。对于该模型，我们在短范围（围绕边界）和长范围（远离边界）范围内为相应的零密度函数找到了精确的渐近律。作为推论，相对于随机球谐函数的旋转不变模型，我们能够找到该集合的总节点长度的对数负偏差。让·布尔加因（Jean Bourgain）的研究以及对拉普拉斯本征函数的节点几何的热情研究，对该领域和当前的趋势产生了至关重要的影响。他关于频谱相关性的工作{Krishnapur等人的工作中的定理2.2 。[安 数学。177（2），699–737（2013）]}和Bombieri和Bourgain [Int。数学。Res。不是。（IMRN）11，3343-3407（2015）]已经打开的门上的上的算术风味表面，如环面或平方定义函数的节长度的有源正在进行的研究。此外，Bourgain的著作[J. 布尔甘，Isr。J.数学。[201（2），611–630（2014）]，也考虑了频谱相关性，可以推论其随机高斯对应物的确定性结果。