Averages of determinants of Laplacians over moduli spaces for large genus
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909918283431936 |
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| author | He, Yuxin Wu, Yunhui |
| author_facet | He, Yuxin Wu, Yunhui |
| contents | Let $\mathcal{M}_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. We view the regularized determinant $\log \det(Δ_{X})$ of Laplacian as a function on $\mathcal{M}_g$ and show that there exists a universal constant $E>0$ such that as $g\to \infty$,
(1) the expected value of $\left|\frac{\log \det(Δ_{X})}{4π(g-1)}-E \right|$ over $\mathcal{M}_g$ has rate of decay $g^{-δ}$ for some uniform constant $δ\in (0,1)$;
(2) the expected value of $\left|\frac{\log \det(Δ_{X})}{4π(g-1)}\right|^β$ over $\mathcal{M}_g$ approaches to $E^β$ whenever $β\in [1,2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12971 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Averages of determinants of Laplacians over moduli spaces for large genus He, Yuxin Wu, Yunhui Geometric Topology Differential Geometry Spectral Theory 32G15, 58J52, 57K20 Let $\mathcal{M}_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. We view the regularized determinant $\log \det(Δ_{X})$ of Laplacian as a function on $\mathcal{M}_g$ and show that there exists a universal constant $E>0$ such that as $g\to \infty$, (1) the expected value of $\left|\frac{\log \det(Δ_{X})}{4π(g-1)}-E \right|$ over $\mathcal{M}_g$ has rate of decay $g^{-δ}$ for some uniform constant $δ\in (0,1)$; (2) the expected value of $\left|\frac{\log \det(Δ_{X})}{4π(g-1)}\right|^β$ over $\mathcal{M}_g$ approaches to $E^β$ whenever $β\in [1,2)$. |
| title | Averages of determinants of Laplacians over moduli spaces for large genus |
| topic | Geometric Topology Differential Geometry Spectral Theory 32G15, 58J52, 57K20 |
| url | https://arxiv.org/abs/2411.12971 |