Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus

@article{Mirzakhani2010GrowthOW,
  title={Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus},
  author={Maryam Mirzakhani},
  journal={arXiv: General Topology},
  year={2010}
}
  • M. Mirzakhani
  • Published 10 December 2010
  • Mathematics
  • arXiv: General Topology
In this paper we study the asymptotic behavior of Weil-Petersson volumes of moduli spaces of hyperbolic surfaces of genus $g$ as $g \rightarrow \infty.$ We apply these asymptotic estimates to study the geometric properties of random hyperbolic surfaces, such as the Cheeger constant and the length of the shortest simple closed geodesic of a given combinatorial type. 
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References

SHOWING 1-10 OF 57 REFERENCES
On Weil-Petersson Volumes and Geometry of Random Hyperbolic Surfaces
This paper investigates the geometric properties of random hyperbolic surfaces with respect to the Weil-Petersson measure. We describe the relationship between the behavior of lengths of simple
Weil–Petersson volumes and cone surfaces
The moduli spaces of hyperbolic surfaces of genus g with n geodesic boundary components are naturally symplectic manifolds. Mirzakhani proved that their volumes are polynomials in the lengths of the
On the large genus asymptotics of Weil-Petersson volumes
A relatively fast algorithm for evaluating Weil-Petersson volumes of moduli spaces of complex algebraic curves is proposed. On the basis of numerical data, a conjectural large genus asymptotics of
Contemporary Mathematics Random hyperbolic surfaces and measured laminations
We prove an equidistribution result for the level sets of the lengths of simple closed curves in the moduli spaceMg of hyperbolic surfaces of genus g. This result parallels known results regarding
On Asymptotic Weil-Petersson Geometry of Teichmuller Space of Riemann Surfaces
We investigate the asymptotic behavior of curvatures of the Weil-Petersson metric in Teichmuller space. We use a pointwise curvature estimate to study directions, in the tangent space, of extremely
Towards large genus asymptotics of intersection numbers on moduli spaces of curves
We explicitly compute the diverging factor in the large genus asymptotics of the Weil–Petersson volumes of the moduli spaces of n-pointed complex algebraic curves. Modulo a universal multiplicative
Higher Weil-Petersson volumes of moduli spaces of stablen-pointed curves
Moduli spaces of compact stablen-pointed curves carry a hierarchy of cohomology classes of top dimension which generalize the Weil-Petersson volume forms and constitute a version of Mumford classes.
THE WEIL-PETERSSON GEOMETRY OF THE MODULI SPACE OF RIEMANN SURFACES
In 2007, Z. Huang showed that in the thick part of the moduli space M g of compact Riemann surfaces of genus g, the sectional curvature of the Weil-Petersson metric is bounded below by a constant
Estimates of Weil–Petersson Volumes¶via Effective Divisors
Abstract: We study the asymptotics of the Weil–Petersson volumes of the moduli spaces of compact Riemann surfaces of genus g with n punctures, for fixed n as g→∞.
Recursion Formulae of Higher Weil–Petersson Volumes
In this paper, we study effective recursion formulae for computing intersection numbers of mixed and classes on moduli spaces of curves. By using the celebrated Witten-Kontsevich theorem, we
...
1
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