Length partition of random multicurves on large genus hyperbolic surfaces

Abstract

We study the length statistics of the components of a random multicurve on a closed surface of genus at least 2. This investigation was initiated by Mirzakhani in a paper published in 2016 where she studied the case of random pants decompositions. We prove that as the genus tends to infinity these statistics converge in law to the Poisson–Dirichlet distribution with parameter 1/2. In particular, the mean lengths of the three longest components converge to 75.8%, 17.1% and 4.9% of the total length, respectively.