Eigenfunctions and Random Waves in the Benjamini-Schramm limit

Abért, Miklós and Bergeron, Nicolas and Le Masson, Etienne (2023) Eigenfunctions and Random Waves in the Benjamini-Schramm limit. Journal of Topology and Analysis. ISSN 1793-5253

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Abstract

We investigate the asymptotic behavior of eigenfunctions of the Laplacian on Riemannian manifolds. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue aspects of the theory. As a result, we present a mathematically precise formulation of Berry’s random wave conjecture for a compact negatively curved manifold and formulate a Berry-type conjecture for sequences of locally symmetric spaces. We prove some weak versions of these conjectures. Using ergodic theory, we also analyze the connections of these conjectures to Quantum Unique Ergodicity.

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