Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves

We express the Masur-Veech volume and the area Siegel-Veech constant of the space of meromorphic quadratic differential with simple poles as in the intersection numbers of psi-classes supported on the cycles of the Deligne-Mumford compactification of the moduli space of. Our formulae are derived from lattice point count involving the volume polynomials that also appear in Mirzakhani's recursion for Weil-Petersson volumes of the moduli space of bordered hyperbolic Riemann. A similar formula for the Masur-Veech volume (though without explicit) was obtained earlier by Mirzakhani through completely different. We prove further result: up to an explicit normalization factor only on the genus and on the number of cusps, the density of the of any simple closed multicurve computed by Mirzakhani coincides with the of square-tiled surfaces having horizontal cylinder decomposition to the simple closed multicurve. We study the resulting densities in more detail in the special case when are no cusps. In particular, we compute explicitly the asymptotic of separating and non-separating simple closed geodesics on a hyperbolic surface of genus g for all small genera g and we show that in genera the separating closed geodesics are exponentially less frequent. We conclude with detailed conjectural description of combinatorial geometry a random simple closed multicurve on a surface of large genus and of a square-tiled surface of large genus. This description is conditional to conjectural asymptotic formula for the Masur-Veech volume in large genera to the conjectural uniform asymptotic formula for certain sums of numbers of psi-classes in large genera.