Mixing and decorrelation in infinite measure: the case of the periodic sinai billiard

We investigate the question of the rate of mixing for observables of a Z d-extension of a probability preserving dynamical system with good spectral properties. We state general mixing results, including expansions of every order. The main part of this article is devoted to the study of mixing rate for smooth observables of the Z 2-periodic Sinai billiard, with different kinds of results depending on whether the horizon is finite or infinite. We establish a first order mixing result when the horizon is infinite. In the finite horizon case, we establish an asymptotic expansion of every order, enabling the study of the mixing rate even for observables with null integrals