Uniform spectral gaps for random hyperbolic surfaces with not many cusps
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912890095665152 |
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| author | He, Yuxin Wu, Yunhui Xue, Yuhao |
| author_facet | He, Yuxin Wu, Yunhui Xue, Yuhao |
| contents | In this paper, we investigate uniform spectral gaps for Weil-Petersson random hyperbolic surfaces with not many cusps. We show that if $n=O(g^α)$ where $α\in \left[0,\frac{1}{2}\right)$, then for any $ε>0$, a random cusped hyperbolic surface in $\mathcal{M}_{g,n}$ has no eigenvalues in $\left(0,\frac{1}{4}-\left(\frac{1}{6(1-α)}\right)^2-ε\right)$. If $α$ is close to $\frac{1}{2}$, this gives a new uniform lower bound $\frac{5}{36}-ε$ for the spectral gaps of Weil-Petersson random hyperbolic surfaces. The major contribution of this work is to reveal a critical phenomenon of ``second order cancellation". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08352 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform spectral gaps for random hyperbolic surfaces with not many cusps He, Yuxin Wu, Yunhui Xue, Yuhao Differential Geometry Complex Variables Geometric Topology Spectral Theory 32G15, 57K20, 58C40 In this paper, we investigate uniform spectral gaps for Weil-Petersson random hyperbolic surfaces with not many cusps. We show that if $n=O(g^α)$ where $α\in \left[0,\frac{1}{2}\right)$, then for any $ε>0$, a random cusped hyperbolic surface in $\mathcal{M}_{g,n}$ has no eigenvalues in $\left(0,\frac{1}{4}-\left(\frac{1}{6(1-α)}\right)^2-ε\right)$. If $α$ is close to $\frac{1}{2}$, this gives a new uniform lower bound $\frac{5}{36}-ε$ for the spectral gaps of Weil-Petersson random hyperbolic surfaces. The major contribution of this work is to reveal a critical phenomenon of ``second order cancellation". |
| title | Uniform spectral gaps for random hyperbolic surfaces with not many cusps |
| topic | Differential Geometry Complex Variables Geometric Topology Spectral Theory 32G15, 57K20, 58C40 |
| url | https://arxiv.org/abs/2602.08352 |