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Large Genus Asymptotic Geometry of Random Square-tiled Surfaces and of Random Multicurves by Delecroix, Vincent; Goujard, Elise; Zograf, Peter; Zorich, Anton is a scholarly article available to read on EtoBox.

What is Large Genus Asymptotic Geometry of Random Square-tiled Surfaces and of Random Multicurves about?

We study the combinatorial geometry of a random closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. We prove that primitive components of a random multicurve represent linearly independent homology cycles with asymptotic probability 1 and that it is primitive with asymptotic probability $\sqrt{2}/2$. We prove analogous properties for random square-tiled surfaces. In particular, we show that all conical singularities of a random square-tiled surface belong to the same leaf of the horizontal foliation and to the same leaf of the vertical foliation with asymptotic probability 1. We show that the number of components of a random multicurve and the number of maximal horizontal cylinders of a random square-tiled surface of genus $g$ are both very well-approximated by the number of cycles of a random permutation for an explicit non-uniform measure on the symmetric group of $3g-3$ elements. In particular, we prove that the expected value of these quantities is asymptotically equivalent to $(\log(6g-6) + \gamma)/2 + \log 2$. These results are based on our formula for the Masur--Veech volume of the moduli space of holomorphic quadratic different

Author

Delecroix, Vincent; Goujard, Elise; Zograf, Peter; Zorich, Anton

Published

2020

Language

EN