Short geodesics and small eigenvalues on random hyperbolic punctured spheres

  • Will Hide

    Durham University, UK
  • Joe Thomas

    Durham University, UK
Short geodesics and small eigenvalues on random hyperbolic punctured spheres cover
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Abstract

We study the number of short geodesics and small eigenvalues on Weil–Petersson random genus zero hyperbolic surfaces with cusps in the regime . Inspired by work of Mirzakhani and Petri (2019), we show that the random multiset of lengths of closed geodesics converges, after a suitable rescaling, to a Poisson point process with explicit intensity. As a consequence, we show that the Weil–Petersson probability that a hyperbolic punctured sphere with cusps has at least arbitrarily small eigenvalues tends to as .

Cite this article

Will Hide, Joe Thomas, Short geodesics and small eigenvalues on random hyperbolic punctured spheres. Comment. Math. Helv. 100 (2025), no. 3, pp. 463–506

DOI 10.4171/CMH/588