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Abstract

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 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 the
Deligne-Mumford compactification of the moduli space of curves proved by
A.~Aggarwal.