Large-$n$ asymptotics for Weil-Petersson volumes of moduli spaces of bordered hyperbolic surfaces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909470356930560 |
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| author | Hide, Will Thomas, Joe |
| author_facet | Hide, Will Thomas, Joe |
| contents | We study the geometry and spectral theory of Weil-Petersson random surfaces with genus-$g$ and $n$ cusps in the large-$n$ limit. We show that for a random hyperbolic surface in $\mathcal{M}_{g,n}$ with $n$ large, the number of small Laplacian eigenvalues is linear in $n$ with high probability. By work of Otal and Rosas [41], this result is optimal up to a multiplicative constant. We also study the relative frequency of simple and non-simple closed geodesics, showing that on random surfaces with many cusps, most closed geodesics with lengths up to $\log(n)$ scales are non-simple. Our main technical contribution is a novel large-$n$ asymptotic formula for the Weil-Petersson volume $V_{g,n}\left(\ell_{1},\dots,\ell_{k}\right)$ of the moduli space $\mathcal{M}_{g,n}\left(\ell_{1},\dots,\ell_{k}\right)$ of genus-$g$ hyperbolic surfaces with $k$ geodesic boundary components and $n-k$ cusps with $k$ fixed, building on work of Manin and Zograf [30]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11412 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Large-$n$ asymptotics for Weil-Petersson volumes of moduli spaces of bordered hyperbolic surfaces Hide, Will Thomas, Joe Geometric Topology Algebraic Geometry Differential Geometry Probability Spectral Theory 58J50, 32G15 We study the geometry and spectral theory of Weil-Petersson random surfaces with genus-$g$ and $n$ cusps in the large-$n$ limit. We show that for a random hyperbolic surface in $\mathcal{M}_{g,n}$ with $n$ large, the number of small Laplacian eigenvalues is linear in $n$ with high probability. By work of Otal and Rosas [41], this result is optimal up to a multiplicative constant. We also study the relative frequency of simple and non-simple closed geodesics, showing that on random surfaces with many cusps, most closed geodesics with lengths up to $\log(n)$ scales are non-simple. Our main technical contribution is a novel large-$n$ asymptotic formula for the Weil-Petersson volume $V_{g,n}\left(\ell_{1},\dots,\ell_{k}\right)$ of the moduli space $\mathcal{M}_{g,n}\left(\ell_{1},\dots,\ell_{k}\right)$ of genus-$g$ hyperbolic surfaces with $k$ geodesic boundary components and $n-k$ cusps with $k$ fixed, building on work of Manin and Zograf [30]. |
| title | Large-$n$ asymptotics for Weil-Petersson volumes of moduli spaces of bordered hyperbolic surfaces |
| topic | Geometric Topology Algebraic Geometry Differential Geometry Probability Spectral Theory 58J50, 32G15 |
| url | https://arxiv.org/abs/2312.11412 |