On the approximate normality of eigenfunctions of the Laplacian

Elizabeth Meckes

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arXiv · 0705.1342

On the approximate normality of eigenfunctions of the Laplacian

Elizabeth Meckes

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Abstract

The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If $X$ is a random point on a manifold $M$ and $f$ is an eigenfunction of the Laplacian with $L^2$-norm one and eigenvalue $-μ$, then $$d_{TV}(f(X),Z)\le\frac{2}μ\E\big|\|\nabla f(X)\|^2-\E\|\nabla f(X) \|^2\big|.$$ This result is applied to construct specific examples of spherical harmonics of arbitrary (odd) degree which are close to Gaussian in distribution. A second application is given to random linear combinations of eigenfunctions on flat tori.

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BibTeX RIS

Elizabeth Meckes. 2007-05-09. On the approximate normality of eigenfunctions of the Laplacian. https://arxiv.org/abs/0705.1342

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