Eigenvalue rigidity of hyperbolic surfaces in the random cover model
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866911545863176192 |
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| author | Kim, Elena Tao, Zhongkai |
| author_facet | Kim, Elena Tao, Zhongkai |
| contents | Let $X$ be a compact connected orientable hyperbolic surface and let $X_n$ be a degree $n$ random cover. We show that, with high probability, the distribution of eigenvalues of the Laplacian on $X_n$ converges to the spectral measure of the hyperbolic plane with polynomially decaying error. This is analogous to the eigenvalue rigidity property for graphs of Huang--Yau [arXiv:2102.00963] and improves the logarithmic bound of Monk [arXiv:2002.00869]. We also obtain a polynomial improvement on the $L^{\infty}$ bound of the eigenfunctions. Our proof relies on the Selberg trace formula and a variant of the polynomial method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01127 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Eigenvalue rigidity of hyperbolic surfaces in the random cover model Kim, Elena Tao, Zhongkai Spectral Theory Geometric Topology Probability Let $X$ be a compact connected orientable hyperbolic surface and let $X_n$ be a degree $n$ random cover. We show that, with high probability, the distribution of eigenvalues of the Laplacian on $X_n$ converges to the spectral measure of the hyperbolic plane with polynomially decaying error. This is analogous to the eigenvalue rigidity property for graphs of Huang--Yau [arXiv:2102.00963] and improves the logarithmic bound of Monk [arXiv:2002.00869]. We also obtain a polynomial improvement on the $L^{\infty}$ bound of the eigenfunctions. Our proof relies on the Selberg trace formula and a variant of the polynomial method. |
| title | Eigenvalue rigidity of hyperbolic surfaces in the random cover model |
| topic | Spectral Theory Geometric Topology Probability |
| url | https://arxiv.org/abs/2603.01127 |