We study monochromatic random waves on $\mathbb{R}^n$ defined by Gaussian variables whose variances tend to zero sufficiently fast. This has the effect that the Fourier transform of the monochromatic wave is an absolutely continuous measure on the sphere with a suitably smooth density, which connects the problem with the scattering regime of monochromatic waves. In this setting, we compute the asymptotic distribution of the nodal components of random monochromatic waves, showing that the number of nodal components contained in a large ball $B_R$ grows asymptotically like $R/\pi$ with probability $p_n>0$, and is bounded uniformly in $R$ with probability $1-p_n$ (which is positive if and only if $n \geq 3$). In the latter case, we show the existence of a unique noncompact nodal component. We also provide an explicit sufficient stability criterion to ascertain when a more general Gaussian probability distribution has the same asymptotic nodal distribution law.

中文翻译：

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非相同分布单色随机波的节点分量的渐近性

我们研究由高斯变量定义的 $\mathbb{R}^n$ 上的单色随机波，其方差趋于足够快地为零。这会导致单色波的傅立叶变换是对具有适当平滑密度的球体的绝对连续测量，这将问题与单色波的散射机制联系起来。在这个设置中，我们计算随机单色波的节点分量的渐近分布，表明包含在一个大球 $B_R$ 中的节点分量的数量像 $R/\pi$ 一样渐近增长，概率为 $p_n>0$，并且在 $R$ 中以 $1-p_n$ 的概率统一有界（当且仅当 $n \geq 3$ 为正）。在后一种情况下，我们展示了一个独特的非紧凑节点组件的存在。