arXiv 2005.063892020-05math.PRmath-phmath.MP

黎曼随机波的几乎必然渐近性

Almost sure asymptotics for Riemannian random waves

Louis Gass

备注：: Revised version (v4). 37 pages

摘要：

考虑一般紧致黎曼流形上Laplace本征函数的高斯线性组合的黎曼随机波模型。与概率1相对于高斯系数，我们建立，无论是大波段和单色模型，适当的重新缩放和评估的过程中，在一个独立的和统一的选择点$X$的流形上，收敛于分布下的唯一随机性的$X$对一个普遍的高斯场的频率趋于无穷大。这一结果将Salem-Zygmund关于三角多项式级数的著名的中心极限定理推广到更一般的紧黎曼流形的框架中。然后，我们推导出上述收敛的几乎必然渐近的节点体积与随机波。据我们所知，在真正的黎曼情形下，由于缺乏足够精确的方差估计，这些渐近性仅在期望中已知，而不是在几乎必然的意义上。这特别解决了S的问题。Zelditch关于节点体积的几乎必然均匀分布。

英文摘要：

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.

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