The periodic wind-tree model is an infinite billiard in the plane with identical rectangular scatterers placed at each integer point. We prove that independtly of the size of scatters and generically with respect to the angle, the polynomial diffusion rate in this billiard is 2/3. Résumé Diffusion du vent dans les arbresLe vent dans les arbres périodique est un billard infini construit de la manière suivante. On considère le plan dans lequel sont placés des obstacles rectangulaires identiques à chaque point entier. Une particule (identifiée à un point) se déplace en ligne droite (le vent) et rebondit de manière élastique sur les obstacles (les arbres). Nous prouvons qu'indépendamment de la taille des obstacles et génériquement par rapport à l'angle initial de la particule le coefficient de diffusion polynomial des orbites de ce billard est 2/3.

No abstract

We express the Masur-Veech volume Vol Qg,n and the area Siegel-Veech constant carea (Qg,n) of the moduli space Qg,n of meromorphic quadratic differential with n simple poles and no other poles as polynomials in the intersection numbers of ψ-classes supported on the boundary cycles of the Deligne-Mumford compactification Mg,n. The formulae obtained in this article are derived from lattice point count involving the Kontsevich volume polynomials Ng,n(b 1 , . . . , bn) that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli space Mg,n(b 1 , . . . , bn) of bordered hyperbolic Riemann surfaces.A similar formula for the Masur-Veech volume Vol Qg (though without explicit evaluation) was obtained earlier by M. Mirzakhani through completely different approach. We prove further result: up to a normalization factor depending only on g and n (which we compute explicitly), the density of the orbit Modg,n •γ of any simple closed multicurve γ inside the ambient set MLg,n(Z) of integral measured laminations computed by Mirzakhani, coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to γ among all square-tiled surfaces in Qg,n.We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case when n = 0. In particular, we compute explicitly the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g and we show that in large genera the separating closed geodesics are 2 3πg • 1 4 g times less frequent.We conclude with detailed conjectural description of combinatorial geometry of a random simple closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. This description is conditional to the conjectural asymptotic formula for the Masur-Veech volume Vol Qg in large genera and to the conjectural uniform asymptotic formula for certain sums of intersection numbers of ψ-classes in large genera.

The periodic wind-tree model is a family T (a, b) of billiards in the plane in which identical rectangular scatterers of size a × b are disposed at each integer point. It was proven by P. Hubert, S. Lelièvre and S. Troubetzkoy that for a residual set of parameters (a, b) the billiard flow in T (a, b) is recurrent in almost every direction. We prove that for many parameters (a, b) there exists a set Λ ⊂ S 1 of positive Hausdorff dimension such that for every θ ∈ Λ every billiard trajectory in T (a, b) with initial angle θ is divergent. Résumé Trajectoires divergentes pour vent dans les arbresLe "vent dans les arbres" est une famille de billards infinis T (a, b) définis de la manière suivante. Dans le plan euclidien R 2 , on place des rectangles de taille a × b à chaque point entier. Une particule (identifiée à un point) se déplace en ligne droite et rebondit de manière élastique sur les obstacles. P. Hubert, S. Lelièvre et S. Troubetzkoy ont démontré qu'il existait un G δ dense de paramètres (a, b) pour lesquels, dans presque toute direction θ ∈ S 1 , le flot du billard T (a, b) dans la direction θ est récurrent. Nous prouvons que pour certains paramètres (a, b), il existe un ensemble Λ ⊂ S 1 de mesure de Hausdorff positive tel que pour tout θ ∈ Λ toute trajectoire dans le billard T (a, b) dont l'angle de départ est θ est divergente.