Delocalisation of eigenfunctions on large genus random surfaces

2020/02/04 by Thomas, Joe · 1 citation
#32G15 (Secondary) #58J50 (Primary) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2002.01403

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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