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

TL;DR

This paper investigates the asymptotic behavior of the number of nodal components of fractional Gaussian fields on Riemannian manifolds, revealing different regimes depending on the parameter s relative to the dimension n.

Contribution

It provides precise asymptotic formulas for the expected number of nodal components of fractional Gaussian fields, including critical and supercritical cases, and bounds for topological invariants.

Findings

For s<n/2, the expected number of nodal components scales as L^{n/2}.

At s=n/2, the expectation involves a logarithmic correction.

For s>n/2, the variance of the field converges, indicating non-universal behavior.

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