Equidistribution of square-tiled surfaces, meanders, and Masur–Veech volumes

We show how recent results of the authors on equidistribution of square-tiled surfaces of given combinatorial type allow to compute approximate values of Masur–Veech volumes of the strata in the moduli spaces of Abelian and quadratic differentials by Monte Carlo method.
We also show how similar approach allows to count asymptotical number of meanders of fixed combinatorial type in various settings in all genera. Our formulae are particularly efficient for classical meanders in genus zero.
We construct a bridge between flat and hyperbolic worlds giving a formula for the Masur–Veech volume of the moduli space of quadratic differentials in terms of intersection numbers of $\mathcal{M}_{g,n}$ (in the spirit of Mirzakhani's formula for Weil–Peterson volume of the moduli space of pointed curves).
Finally we present several conjectures concerning Masur–Veech volumes.