Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves

Abstract

We study the combinatorial geometry of a random closed multicurve on a surface of large genus g and of a random square-tiled surface of large genus g. We prove that primitive components γ 1 , . . . , γ k of a random multicurve m 1 γ 1 + • • • + m k γ k represent linearly independent homology cycles with asymptotic probability 1 and that all its weights m i are equal to 1 with asymptotic probability √ 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) + γ)/2 + log 2.These results are based on our formula for the Masur-Veech volume Vol Qg of the moduli space of holomorphic quadratic differentials combined with deep large genus asymptotic analysis of this formula performed by A. Aggarwal and with the uniform asymptotic formula for intersection numbers of ψ-classes on Mg,n for large g proved by A. Aggarwal in 2020.

Reconstructing GKZ via Topological Recursion

“…In this approach, the combinatorial data of the distribution of simple closed geodesics is described by stable graphs, and the Masur-Veech volumes are computed by combinations of Weil-Petersson volumes. In recent years, Mirzakhani's combinatorial approach to compute the Masur-Veech volumes is established further in a series of works by Delecroix, Goujard, Zograf, and Zorich [21,22,23,24,25]. (See also [44,17] for the related works.)…”

Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves

“…A conjectural generalization of Formula (1.17) to other strata of meromorphic quadratic differentials and numerical evidence beyond this conjecture are presented in [ADGZZ]. The statistical geometry of random square-tiled surfaces of large genus and of random simple closed multicurves on surfaces of large genus is discussed in the separate paper [DGZZ5].…”

“…Remark 1.18. In order to go beyond the case of simple closed curve, one has to carry a much more involved asymptotic analysis of correlators that we perform in full generality in [DGZZ5]. Example 1.19.…”