Short geodesic loops and $$L^p$$ norms of eigenfunctions on large genus random surfaces

Gilmore, Clifford; Masson, E.; Sahlsten, Tuomas; Thomas, Joe

doi:10.1007/s00039-021-00556-6

Cited by 15 publications

(10 citation statements)

References 42 publications

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“…The purpose of this appendix is to prove the necessary Weil-Petersson volume estimates used in the proof of Theorem 4.1. Similar estimates can be found in, for example, [5,14,16,17,25].…”

Section: A Volume Estimatessupporting

confidence: 80%

See 1 more Smart Citation

Spectral Gap for Weil–Petersson Random Surfaces with Cusps

Will

2022

International Mathematics Research Notices

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We show that for any $\varepsilon>0$, $\alpha \in [0,\frac {1}{2})$, as $g\to \infty $ a generic finite-area genus $g$ hyperbolic surface with $n=O\left (g^{\alpha }\right )$ cusps, sampled with probability arising from the Weil–Petersson metric on moduli space, has no non-zero eigenvalue of the Laplacian below $\frac {1}{4}-\left (\frac {2\alpha +1}{4}\right )^{2}-\varepsilon $. For $\alpha =0$ this gives a spectral gap of size $\frac {3}{16}-\varepsilon $ and for any $\alpha <\frac {1}{2}$ gives a uniform spectral gap of explicit size.

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“…The purpose of this appendix is to prove the necessary Weil-Petersson volume estimates used in the proof of Theorem 4.1. Similar estimates can be found in, for example, [5,14,16,17,25].…”

Section: A Volume Estimatessupporting

confidence: 80%

“…The work of Monk in [18] gives estimates on the density of Laplace eigenvalues below 1 4 for Weil-Petersson random compact surfaces. In [5], Gilmore, Le Masson, Sahlsten, and Thomas obtain bounds for the L p norms of Laplace eigenfunctions for Weil-Petersson random compact surfaces.…”

Section: Spectral Theory In the Weil-petersson Modelmentioning

confidence: 99%

Spectral Gap for Weil–Petersson Random Surfaces with Cusps

Will

2022

International Mathematics Research Notices

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“…In this paper, we view certain geometric quantities as random variables on

\mathcal {M}_g

, and study their asymptotic behaviors as

g\rightarrow \infty

. One may also see [9–11, 23, 24] for related interesting topics.…”

Section: Preliminariesmentioning

confidence: 99%

Large genus asymptotics for lengths of separating closed geodesics on random surfaces

Nie

Xue

2023

Journal of Topology

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In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus g with respect to the Weil-Petersson measure on the moduli space  g . We show that as g goes to infinity, a generic surface 𝑋 ∈  g satisfies asymptotically:(1) the separating systole of 𝑋 is approximately 2 log g;(2) there is a half-collar of width approximately log g 2 around any separating systolic curve on 𝑋;(3) the length of the shortest separating closed multigeodesics on 𝑋 is approximately 2 log g.As applications, we also discuss the asymptotic behavior of the extremal separating systole, the non-simple systole, and the expected length of the shortest separating closed multi-geodesics as g goes to infinity.

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“…The work of Monk in [Mo21] gives estimates on the density of Laplace eigenvalues below 1 4 for Weil-Petersson random compact surfaces. In [GMST21], Gilmore, Le Masson, Sahlsten and Thomas obtain bounds for the L p norm of Laplace eigenfunctions for Weil-Petersson random compact surfaces.…”

Section: Introductionmentioning

confidence: 99%

Spectral gap for Weil-Petersson random surfaces with cusps

Will¹

2021

Preprint

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We show that for any > 0, α ∈ [0, 1 2 ), as g → ∞ a generic finite area genus g hyperbolic surface with n = O(g α ) cusps, sampled with probability arising from the Weil-Petersson metric, has no non-zero eigenvalue of the Laplacian below 1 4 − 17 8 α+1 1 4 α+4 2 − . For α = 0 this gives a spectral gap of size 3 16 − and for any α < 1 2 gives a uniform spectral gap of explicit size.

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Short geodesic loops and $$L^p$$ norms of eigenfunctions on large genus random surfaces

Cited by 15 publications

References 42 publications

Spectral Gap for Weil–Petersson Random Surfaces with Cusps

Spectral Gap for Weil–Petersson Random Surfaces with Cusps

Large genus asymptotics for lengths of separating closed geodesics on random surfaces

Spectral gap for Weil-Petersson random surfaces with cusps

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