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We study the combinatorial geometry of a random closed multicurve on asurface of large genus and of a random square-tiled surface of large genus. Weprove that primitive components of a random multicurve represent linearlyindependent homology cycles with asymptotic probability 1 and that it isprimitive with asymptotic probability $\sqrt{2}/2$. We prove analogousproperties for random square-tiled surfaces. In particular, we show that allconical singularities of a random square-tiled surface belong to the same leafof the horizontal foliation and to the same leaf of the vertical foliation withasymptotic probability 1.  We show that the number of components of a random multicurve and the numberof 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 permutationfor an explicit non-uniform measure on the symmetric group of $3g-3$ elements.In particular, we prove that the expected value of these quantities isasymptotically equivalent to $(\log(6g-6) + \gamma)/2 + \log 2$.  These results are based on our formula for the Masur--Veech volume of themoduli space of holomorphic quadratic differentials combined with deep largegenus asymptotic analysis of this formula performed by A.~Aggarwal and with theuniform asymptotic formula for intersection numbers of $\psi$-classes on theDeligne-Mumford compactification of the moduli space of curves proved byA.~Aggarwal.

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