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PreprintCC BY 4.0arXiv

Spectral gaps near one quarter for Weil--Petersson random surfaces with growing cusp number

  • Guan Huang
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Abstract

We prove that the spectral gap of a Weil--Petersson random hyperbolic surface of genus g with n=n(g)≥1 labelled cusps converges in probability to 1/4 as g→∞, for every sequence n=o(√g). More precisely, for each fixed 0<ε<1/4, there are constants Cε≥2 and gε≥3, independent of n, such that the expected number of nonzero L2 eigenvalues in (0,1/4−ε], counted with multiplicity and including residual eigenvalues, is at most Cε(n+1)2/g whenever g≥ gε and g≥ Cε(n+1)2. This proves a conjecture of Shen and Wu. A matching lower bound shows that the count estimate is optimal up to constants for each fixed ε, and that n=o(√g) is necessary for convergence of the gap to 1/4. The proof combines volume estimates uniform in the cusp count with polynomial approximation and a positive trace on truncated surfaces.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026spectral,
  title = {{Spectral gaps near one quarter for Weil--Petersson random surfaces with growing cusp number}},
  author = {Guan Huang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01477},
  eprint = {2611.01477},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01477},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Spectral gaps near one quarter for Weil--Petersson random surfaces with growing cusp number
AU  - Guan Huang
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01477
UR  - https://arxiv.org/abs/2611.01477
AB  - We prove that the spectral gap of a Weil--Petersson random hyperbolic surface of genus g with n=n(g)≥1 labelled cusps converges in probability to 1/4 as g→∞, for every sequence n=o(√g). More precisely, for each fixed 0<ε<1/4, there are constants C_ε≥2 and g_ε≥3, independent of n, such that the expected number of nonzero L^2 eigenvalues in (0,1/4−ε], counted with multiplicity and including residual eigenvalues, is at most C_ε(n+1)^2/g whenever g≥ g_ε and g≥ C_ε(n+1)^2. This proves a conjecture of Shen and Wu. A matching lower bound shows that the count estimate is optimal up to constants for each fixed ε, and that n=o(√g) is necessary for convergence of the gap to 1/4. The proof combines volume estimates uniform in the cusp count with polynomial approximation and a positive trace on truncated surfaces.
ER  -