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<p>We study smooth linear spectral statistics of twisted Laplacians on random <i>n</i>-covers of a fixed compact hyperbolic surface <i>X</i>. We consider two aspects of such statistics. The first is the fluctuations of such statistics in a small energy window around a fixed energy level when averaged over the space of all degree <i>n</i> covers of <i>X</i>. The second is the centered energy variance of a typical surface, a quantity similar to the normal energy variance.</p><p>In the first case, we show a central limit theorem. Specifically, we show that the distribution of such fluctuations tends to a Gaussian with variance given by the corresponding quantity for the Gaussian Orthogonal/Unitary Ensemble (GOE/GUE). In the second case, we show that the centered energy variance of a typical random <i>n</i>-cover is that of the GOE/GUE. In both cases, we consider a double limit where first we let <i>n</i>—the covering degree—go to ∞ then let <i>L</i> → ∞ where 1/<i>L</i> is the window length.</p><p>A fundamental component of our proofs are the results we prove in [11] which concern the random cover model for random surfaces.</p>

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Smooth linear eigenvalue statistics on random covers of compact hyperbolic surfaces—a central limit theorem and almost sure RMT statistics

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  • 发表日期：

    2026年4月6日

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  • Yotam Maoz

摘要

We study smooth linear spectral statistics of twisted Laplacians on random n-covers of a fixed compact hyperbolic surface X. We consider two aspects of such statistics. The first is the fluctuations of such statistics in a small energy window around a fixed energy level when averaged over the space of all degree n covers of X. The second is the centered energy variance of a typical surface, a quantity similar to the normal energy variance.

In the first case, we show a central limit theorem. Specifically, we show that the distribution of such fluctuations tends to a Gaussian with variance given by the corresponding quantity for the Gaussian Orthogonal/Unitary Ensemble (GOE/GUE). In the second case, we show that the centered energy variance of a typical random n-cover is that of the GOE/GUE. In both cases, we consider a double limit where first we let n—the covering degree—go to ∞ then let L → ∞ where 1/L is the window length.

A fundamental component of our proofs are the results we prove in [11] which concern the random cover model for random surfaces.