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Eigenfunctions and Random Waves in the Benjamini-Schramm limit

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arxiv 1810.05601 v2 pith:WBRE33R7 submitted 2018-10-12 math.SP math-phmath.DSmath.MPmath.PR

Eigenfunctions and Random Waves in the Benjamini-Schramm limit

classification math.SP math-phmath.DSmath.MPmath.PR
keywords benjamini-schrammconjectureconjectureseigenfunctionstheoryanalyzeaspectsasymptotic
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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 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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