Length statistics of random multicurves on closed hyperbolic surfaces

Abstract

In this paper, we determine the distribution of the length partition of a random multicurve of fixed topological type on a closed hyperbolic surface using the methods of Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres. This distribution admits a polynomial density, whose coefficients can be expressed explicitly in terms of intersection numbers of psi-classes on the Deligne--Mumford compactification, and in particular it does not depend on the hyperbolic metric. This result generalizes prior work of M. Mirzakhani in the case of random pants decompositions. Results very close to ours are obtained independently and simultaneously by F. Arana-Herrera.