NDLI: Universal Components of Random Nodal Sets

Content Provider	Springer Nature Link
Author	Gayet, Damien Welschinger, Jean Yves
Copyright Year	2016
Abstract	We give, as L grows to infinity, an explicit lower bound of order $${L^{\frac{n}{m}}}$$ 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 $${\Sigma}$$ 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 $${L^{-\frac{1}{m}}}$$ centered at x. These results apply in particular to Laplace–Beltrami and Dirichlet-to-Neumann operators.
Starting Page	777
Ending Page	797
Page Count	21
File Format	PDF
ISSN	00103616
Journal	Communications in Mathematical Physics
Volume Number	347
Issue Number	3
e-ISSN	14320916
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-03-25
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Theoretical, Mathematical and Computational Physics Mathematical Physics Quantum Physics Complex Systems Classical and Quantum Gravitation, Relativity Theory
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Mathematical Physics

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