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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.04584 |
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| _version_ | 1866910437317017600 |
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| author | Hide, Will |
| author_facet | Hide, Will |
| contents | Let $X$ be a finite-area non-compact hyperbolic surface. We study the spectrum of the Laplacian on random covering surfaces of X and on random unitary bundles over X. We show that there is a constant $c > 0$ such that, with probability tending to 1 as $n \to \infty$, a uniformly random degree-$n$ Riemannian covering surface $X_n$ of $X$ has no Laplacian eigenvalues below $\frac{1}{4}-c\frac{(\log\log\log n)^2}{\log \log n}$ other than those of $X$ and with the same multiplicities. We also show that with probability tending to 1 as $n\to \infty$, a random unitary bundle $E_ϕ$ over $X$ of rank $n$ has no Laplacian eigenvalues below $\frac{1}{4}-c\frac{(\log\log n)^2}{\log n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_04584 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Effective lower bounds for spectra of random covers and random unitary bundles Hide, Will Spectral Theory Differential Geometry 58J50, 05C50 Let $X$ be a finite-area non-compact hyperbolic surface. We study the spectrum of the Laplacian on random covering surfaces of X and on random unitary bundles over X. We show that there is a constant $c > 0$ such that, with probability tending to 1 as $n \to \infty$, a uniformly random degree-$n$ Riemannian covering surface $X_n$ of $X$ has no Laplacian eigenvalues below $\frac{1}{4}-c\frac{(\log\log\log n)^2}{\log \log n}$ other than those of $X$ and with the same multiplicities. We also show that with probability tending to 1 as $n\to \infty$, a random unitary bundle $E_ϕ$ over $X$ of rank $n$ has no Laplacian eigenvalues below $\frac{1}{4}-c\frac{(\log\log n)^2}{\log n}$. |
| title | Effective lower bounds for spectra of random covers and random unitary bundles |
| topic | Spectral Theory Differential Geometry 58J50, 05C50 |
| url | https://arxiv.org/abs/2305.04584 |