  We show that any real number in [0,1) is a diffusion rate for the wind-tree
model with rational parameters. We will also provide a criterion in order to
describe the shape of the Lyapunov spectrum of cocycles obtained as suspension
of a representation. As an application, we exhibit an infinite family of
wind-tree billiards for which the interior of the Lyapunov spectrum is a big as
possible: this is the full square (0,1)^2. To the best of the knowledge of the
authors, these are the first complete descriptions where the interior of the
Lyapunov spectrum is known explicitly in dimension two, even for general
Fuchsian groups.
