Length statistics of random multicurves on closed hyperbolic surfaces

Abstract

In this note, we determine the length distribution of components of a random multicurve on a fixed hyperbolic surface using Mirzakhani's equidistribution theorem for horospheres and Margulis' thickening technique. We obtain an explicit formula for the resulting lengths statistics and prove, in particular, that it depends only on the topological type of the multicurve and does not depend on the hyperbolic metric. This result generalizes prior result of M. Mirzakhani, providing the length statistics for multicurves decomposing the surface into pairs of pants. Results very close to ours are obtained independently and simultaneously by F. Arana-Herrera.

Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves

“…, γ k ) with 1 ≤ k ≤ 3g − 3 be an ordered simple closed multi-curve on S g . Then, the following asymptotic formula holds, Theorem 6.7 was proved independently by the author [Ara19a] and Liu [Liu19] using general averaging and unfolding methods introduced by Margulis in his thesis [Mar70]. In [Ara19a] a generalization of Theorem 6.7 for ordered filling closed multi-curves was proved using techniques introduced by Mirzakhani in [Mir16].…”

Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow

“…Counting problems for square-tiled surfaces/curves on hyperbolic surfaces are intricately related to the equidistribution of L-level sets for the intersection number with/hyperbolic length of laminations as one takes L → ∞. When λ is a multicurve, the equidistribution of such "expanding horospheres" to the Masur-Veech measure on the principal stratum of Q 1 M g / the pullback by O of this measure on P 1 M g (sometimes called Mirzakhani measure) was established in [Mir07,AH20b,Liu20] using the geometry of the (symmetrized) Lipschitz metric, the non-divergence of the earthquake flow, and a no-escapeof-mass argument. On the other end of the spectrum, the equidistribution of expanding horospheres for maximal λ to Q 1 M g can be proven using a standard "thickening plus mixing" argument from homogeneous dynamics; in the flat setting this is implicit in the work of Lindenstrauss and Mirzakhani [LM08], and was recently generalized in [For20, Theorem 1.6] using different methods.…”