Universal components of random nodal sets

TL;DR

This paper establishes lower bounds on the expected Betti numbers of random nodal sets of eigenfunctions of elliptic operators on manifolds, showing that certain topological features appear with positive probability as eigenvalues grow.

Contribution

It provides explicit probabilistic lower bounds for the topology of random eigenfunction nodal sets, extending understanding of their universal components across different operators.

Findings

Expected Betti numbers grow at least as fast as L^{n/m}

Components diffeomorphic to any hypersurface appear with positive probability

Results apply to Laplace-Beltrami and Dirichlet-to-Neumann operators

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.