Smooth linear eigenvalue statistics on random covers of compact hyperbolic surfaces -- A central limit theorem and almost sure RMT statistics
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913787761655808 |
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| author | Maoz, Yotam |
| author_facet | Maoz, Yotam |
| contents | 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 energy variance of a typical surface.
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 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 $\infty$ then let $L\to \infty$ where $1/L$ is the window length. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_18663 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Smooth linear eigenvalue statistics on random covers of compact hyperbolic surfaces -- A central limit theorem and almost sure RMT statistics Maoz, Yotam Spectral Theory Mathematical Physics Dynamical Systems Geometric Topology Number Theory Probability 81Q50 (Primary) 11F72, 58J50 (Secondary) 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 energy variance of a typical surface. 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 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 $\infty$ then let $L\to \infty$ where $1/L$ is the window length. |
| title | Smooth linear eigenvalue statistics on random covers of compact hyperbolic surfaces -- A central limit theorem and almost sure RMT statistics |
| topic | Spectral Theory Mathematical Physics Dynamical Systems Geometric Topology Number Theory Probability 81Q50 (Primary) 11F72, 58J50 (Secondary) |
| url | https://arxiv.org/abs/2310.18663 |