论文标题
几乎确定Riemannian随机波的渐近学
Almost sure asymptotics for Riemannian random waves
论文作者
论文摘要
我们考虑了一般紧凑的riemannian歧管上拉普拉斯本征函数高斯线性组合的riemannian随机波模型。对于高斯系数的概率，我们确定，对于大型带和单色模型，该过程在歧管上以独立且均匀选择的点$ x $进行了正确重新缩放和评估，在频率上趋于频率倾向于无限的通用高斯磁场，以$ x $的唯一随机性收敛于分布。该结果将三角多项式系列的Salem-Zygmund的著名中心限制延伸到紧凑的Riemannian歧管的更一般框架。然后，我们从上面的收敛中推断出与随机波相关的节点体积的几乎纯净的渐近物。据我们所知，在实际的riemannian案例中，这些渐近学仅在预期中才知道，而在几乎确定的意义上，由于缺乏足够准确的差异估计。这特别是解决了S. Zelditch的问题，内容涉及淋巴结量的几乎肯定的等分。
We consider the Riemannian random wave model of Gaussian linear combinations of Laplace eigenfunctions on a general compact Riemannian manifold. With probability one with respect to the Gaussian coefficients, we establish that, both for large band and monochromatic models, the process properly rescaled and evaluated at an independently and uniformly chosen point $X$ on the manifold, converges in distribution under the sole randomness of $X$ towards an universal Gaussian field as the frequency tends to infinity. This result extends the celebrated central limit Theorem of Salem--Zygmund for trigonometric polynomials series to the more general framework of compact Riemannian manifolds. We then deduce from the above convergence the almost-sure asymptotics of the nodal volume associated with the random wave. To the best of our knowledge, in the real Riemannian case, these asymptotics were only known in expectation and not in the almost sure sense due to the lack of sufficiently accurate variance estimates. This in particular addresses a question of S. Zelditch regarding the almost sure equidistribution of nodal volume.
