Random lattices, punctured tori and the Teichmüller distribution
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- by Gaven Martin PDF
- Proc. Amer. Math. Soc. 148 (2020), 289-300 Request permission
Abstract:
The moduli space of lattices of $\mathbb {C}$ is a Riemann surface of finite hyperbolic area with the square lattice as an origin. We select a lattice from the induced uniform distribution and calculate the statistics of the Teichmüller distance to the origin. This in turn identifies distribution of the distance in Teichmüller space to the central “square” punctured torus in the moduli space of punctured tori. There are singularities in this p.d.f. arising from the topology of the moduli space. We also consider the statistics of the distance in Teichmüller space to the rectangular punctured tori and the p.d.f. and expected distortion of the extremal quasiconformal mappings.References
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Additional Information
- Gaven Martin
- Affiliation: Institute for Advanced Study, Massey University, Auckland, 0632, New Zealand
- MR Author ID: 120465
- Email: g.j.martin@massey.ac.nz
- Received by editor(s): July 30, 2018
- Received by editor(s) in revised form: April 28, 2019
- Published electronically: July 30, 2019
- Additional Notes: The author’s research was supported in part by grants from the N. Z. Marsden Fund.
- Communicated by: Jeremy Tyson
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 289-300
- MSC (2010): Primary 30C62, 30F40, 30D50, 20H10, 22E40, 53A35, 57N13, 57M60
- DOI: https://doi.org/10.1090/proc/14687
- MathSciNet review: 4042851