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PreprintCC BY-NC-ND 4.0arXiv

Lengths of closed geodesics on random non-compact hyperbolic surfaces

  • Malo Gatignol
  • Joe Thomas
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Abstract

We study the asymptotic distribution of the number of primitive, closed geodesics on Weil-Petersson random hyperbolic surfaces with genus g and n(g) cusps, under various growth rates of n(g) as g→∞. We prove that the limiting distribution is Poisson with parameter depending explicitly on the limit of n(g)2/g. This work is inspired by that of Mirzakhani and Petri's on closed surfaces and Hide and Thomas when g is fixed. Our proof relies on first order expansions for Weil-Petersson moduli space volumes in the n(g)/g→ 0 regime. We also obtain some new existence results for small Laplacian eigenvalues.

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Cite

BibTeXRIS
BibTeX
@misc{gatignol2026lengths,
  title = {{Lengths of closed geodesics on random non-compact hyperbolic surfaces}},
  author = {Malo Gatignol and Joe Thomas},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41002},
  eprint = {2610.41002},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41002},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Lengths of closed geodesics on random non-compact hyperbolic surfaces
AU  - Malo Gatignol
AU  - Joe Thomas
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.41002
UR  - https://arxiv.org/abs/2610.41002
AB  - We study the asymptotic distribution of the number of primitive, closed geodesics on Weil-Petersson random hyperbolic surfaces with genus g and n(g) cusps, under various growth rates of n(g) as g→∞. We prove that the limiting distribution is Poisson with parameter depending explicitly on the limit of n(g)^2/g. This work is inspired by that of Mirzakhani and Petri's on closed surfaces and Hide and Thomas when g is fixed. Our proof relies on first order expansions for Weil-Petersson moduli space volumes in the n(g)/g→ 0 regime. We also obtain some new existence results for small Laplacian eigenvalues.
ER  -