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

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