Universal components of random nodal sets
Résumé
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 randomlinear combination of eigenvectors of $P$ with eigenvalues below $L$. Here, $P$ denotesan elliptic self-adjoint pseudo-differential operatorof order $m>0$, 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 everyclosed 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 $\Sigma$ appearswith probability at least $p_\Sigma$ in the vanishing locus of a random section andin the ball of radius $L^{-1/m}$ centered at $x$. These results apply in particular to Laplace-Beltrami and Dirichlet-to-Neumann operators.
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