Universal components of random nodal sets

Abstract

We give, as $L$ grows to infinity, an explicit lower bound of order $L^{n/m}$ for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of $P$ with eigenvalues below $L$. Here, $P$ denotes an elliptic self-adjoint pseudo-differential operator of order $m\textgreater{}0$, bounded from below and acting on the sections of a Riemannian line bundle over a smooth closed $n$-dimensional manifold $M$ equipped with some Lebesgue measure. In fact, for every closed hypersurface $\Sigma$ of ${\mathbb{R}}^n$, we prove that there exists a positive constant $p\_\Sigma$ depending only on $\Sigma$, such that for every large enough $L$ and every $x\in M$, a component diffeomorphic to $\Sigma$ appears with probability at least $p\_\Sigma$ in the vanishing locus of a random section and in the ball of radius $L^{-1/m}$ centered at $x$. These results apply in particular to Laplace-Beltrami and Dirichlet-to-Neumann operators.

References in corpus (4)

• Betti numbers of random real hypersurfaces and determinants of random symmetric matrices
• Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions
• Statistics on Hilbert's Sixteenth Problem
• On the topology of the zero sets of monochromatic random waves

Cited by in corpus (9)

• Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions
• Eigenfunctions and Random Waves in the Benjamini-Schramm limit
• Tilings, packings and expected Betti numbers in simplicial complexes
• Expected number of nodal components for cut-off fractional Gaussian fields
• Asymptotics for the nodal components of non-identically distributed monochromatic random waves
• Asymptotic topology of random subcomplexes in a finite simplicial complex
• Direction distribution for nodal components of random band-limited functions on surfaces
• Anomalies in local Weyl laws and applications to random topology at critical dimension
• Random Colorings in Manifolds