Cries and whispers in wind-tree forests

We study billiard in the plane endowed with symmetric $\mathbb{Z}^2$-periodic obstacles of a right-angled polygonal shape. One of our main interests is the dependence of the diffusion rate of the billiard on the shape of the obstacle. We prove, in particular, that when the number of angles of a symmetric connected obstacle grows, the diffusion rate tends to zero, thus answering a question of J.-C. Yoccoz. Our results are based on computation of Lyapunov exponents of the Hodge bundle over hyperelliptic loci in the moduli spaces of quadratic differentials, which represents independent interest. In particular, we compute the exact value of the Lyapunov exponent $\lambda^+_1$ for all elliptic loci of quadratic differentials with simple zeroes and poles.