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

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$$f^s\_L=\sum\_{0<\lambda\_j\leq L}\lambda\_j^{-\frac{s}{2}}\xi\_je\_j\, .$$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 $L\rightarrow+\infty$.