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

• 成果类型: Article
• 署名作者: Delecroix, Vincent; Goujard, Elise; Zograf, Peter; Zorich, Anton
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite de Bordeaux; Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Universite de Bordeaux; Centre National de la Recherche Scientifique (CNRS); Inria; Universite Paris Cite
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01123-y
• 发表日期: 2022
• 页码: 123-224
• 关键词: closed geodesics; SCALING LIMITS; MODULI SPACES; Poisson; GROWTH; quadrangulations; transformations; volumes; numbers; CURVES

摘要

We study the combinatorial geometry of a random closed multi-curve on a surface of large genus g and of a random square-tiled surface of large genus g. We prove that primitive components , gamma(1), ...,gamma(k) of a random multicurve m(1)gamma(1) + ... + m(k)gamma(k) represent linearly independent homology cycles with asymptotic probability 1 and that all its weights m(i); are equal to 1 with asymptotic probability root 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 has asymptotics (log(6g - 6) + gamma)/2 + log 2 as g -> infinity, where gamma is the Euler-Mascheroni constant. These results are based on our formula for the Masur-Veech volume Vol Q(g) 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 psi-classes on (M) over bar (g,n) for large g proved by A. Aggarwal in 2020.