Expected number of nodal components for cut-off fractional Gaussian fields

Abstract

Let $(\mathcal{X},g)$ be a closed Riemmanian manifold of dimension $n>0$. Let $\Delta$ be the Laplacian on $\mathcal{X}$, and let $(e\_k)\_k$ be an $L^2$-orthonormal and dense family of Laplace eigenfunctions with respective eigenvalues $(\lambda \_k)\_k$. We assume that $(\lambda \_k)\_k$ is non-decreasing and that the $e\_k$ are real-valued. Let $(\xi \_k)\_k$ be a sequence of iid $\mathcal{N}(0,1)$ random variables. For each $L>0$ and $s\in \mathbb{R}$, possibly negative, set\[f^s\_L=\sum\_{0<λ\_j\leq L}λ\_j^{-\frac{s}{2}}ξ\_je\_j\, .\]Then, $f\_L^s$ is almost surely regular on its zero set. Let $N\_L$ be the number of connected components of its zero set. If $s<\frac{n}{2}$, then we deduce that there exists $\nu =\nu (n,s)>0$ such that $N\_L\sim \nu Vol\_g(\mathcal{X})L^{n/2}$ in $L^1$ and almost surely. In particular, $\mathbb{E}[N\_L]\asymp L^{n/2}$. On the other hand, we prove that if $s=\frac{n}{2}$ then\[{\mathbb{E}}[N\_L]\asymp \frac{L^{n/2}}{\sqrt{\ln\left(L^{1/2}\right)}}\, .\]In the latter case, we also obtain an upper bound for the expected Euler characteristic of the zero set of $f\_L^s$ and for its Betti numbers. In the case $s>n/2$, the pointwise variance of $f\_L^s$ converges so it is not expected to have universal behavior as $L\to +\infty$.

31 pages, corrections of typographical mistakes, correction in Theorem 1.3 and details added in proofs

References in corpus (1)

• Universal components of random nodal sets

Cited by in corpus (2)

• M-dissipative boundary conditions and boundary tuples for Maxwell operators
• Asymptotics for the nodal components of non-identically distributed monochromatic random waves