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.
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Acknowledgements
The author would like to thank Etienne Le Masson and Tuomas Sahlsten for helpful discussions regarding the results presented here. In addition, the author is grateful to Nalini Anantharaman and Laura Monk, as well as Université de Strasbourg for the hospitality during a research visit in February 2020, and for useful comments leading to an improvement on an earlier version of this work. The author also extends thanks to the anonymous referee, whose suggestions improved the presentation of the results.
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Supported by the Dean’s Award from the University of Manchester.
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Thomas, J. Delocalisation of eigenfunctions on large genus random surfaces. Isr. J. Math. 250, 53–83 (2022). https://doi.org/10.1007/s11856-022-2327-1
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DOI: https://doi.org/10.1007/s11856-022-2327-1