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 > 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 of R n , we prove that there exists a positive constant p depending only on , such that for every large enough L and every x ∈ M, a component diffeomorphic to appears with probability at least p 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.

Asymptotic Topology of Random Subcomplexes in a Finite Simplicial Complex

“…These results thus provide counterparts in this combinatorial framework to the ones obtained in [7] and [6,8] for the expected Betti numbers of real algebraic submanifolds of real projective manifolds or nodal domains in smooth manifolds respectively. The paper ends with an appendix devoted to a further study and interpretation of the constants c + i (n, k).…”

Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions

“…Some lower bounds for the number of connected components of the zero set and for other similar quantities were obtained in different settings by Bourgain and Rudnick [7], Fyodorov, Lerario, Lundberg [9], Gayet and Welschinger [10,11,12], Lerario and Lundberg [22] using the "barrier construction" from [25].…”

Hole probability for nodal sets of the cut-off Gaussian Free Field

“…For the lower bound, [6] uses an entropy inequality and the capacity appears by discrete approximation. In our case (see Proposition 16) it seemed more natural to apply the barrier method, already used in [19] and [14]. The idea is to decompose the field into a random multiple of a function h that is greater than one on D and an independent fluctuation.…”