Delocalisation of eigenfunctions on large genus random surfaces

Thomas, Joe

Abstract

We prove that eigenfunctions of the Laplacian on a compact hyperbolic surface delocalise in terms of a geometric parameter dependent upon the number of short closed geodesics on the surface. In particular, we show that an L2 normalised eigenfunction restricted to a measurable subset of the surface has squared L2-norm ε > 0, only if the set has a relatively large size—exponential in the geometric parameter. For random surfaces with respect to the Weil—Petersson probability measure, we then show, with high probability as g → ∞, that the size of the set must be at least the genus of the surface to some power dependent upon the eigenvalue and ε.

Citation

Thomas, J. (2022). Delocalisation of eigenfunctions on large genus random surfaces. Israel Journal of Mathematics, 250(1), 53 -83. https://doi.org/10.1007/s11856-022-2327-1

• Journal Article Type: Article
• Online Publication Date: Jun 29, 2022
• Publication Date: 2022-10
• Deposit Date: Jul 29, 2022
• Publicly Available Date: Jun 29, 2023
• Journal: Israel Journal of Mathematics
• Print ISSN: 0021-2172
• Electronic ISSN: 1565-8511
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 250
• Issue: 1
• Pages: 53 -83
• DOI: https://doi.org/10.1007/s11856-022-2327-1
• Public URL: https://durham-repository.worktribe.com/output/1199133