On the Central Limit Theorem for linear eigenvalue statistics on random surfaces of large genus

2023/01/02 by Zeév Rudnick, Rudnick, Zeév, Igor Wigman +1 · 1 citation
Mathematics · #30F60 #58J50 (Secondary) #81Q50 (Primary) 11F72 #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Number Theory (math.NT) #Spectral Theory (math.SP) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2301.00685

openalex publication_date 2023/01/02 · openalex created_date 2023/01/06 · openalex updated_date 2026/07/30

Abstract

We study the fluctuations of smooth linear statistics of Laplace eigenvalues of compact hyperbolic surfaces lying in short energy windows, when averaged over the moduli space of surfaces of a given genus. The average is taken with respect to the Weil-Petersson measure. We show that first taking the large genus limit, then a short window limit, the distribution tends to a Gaussian. The variance was recently shown to be given by the corresponding quantity for the Gaussian Orthogonal Ensemble (GOE), and the Gaussian fluctuations are also consistent with those in Random Matrix Theory, as conjectured in the physics literature for a fixed surface.

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