Strong mixing for the periodic Lorentz gas flow with infinite horizon

Abstract

We establish strong mixing for the $\mathbb Z^d$-periodic, infinite horizon, Lorentz gas flow for continuous observables with compact support. The essential feature of this natural class of observables is that their support may contain points with infinite free flights. Dealing with such a class of functions is a serious challenge and there is no analogue of it in the finite horizon case. The mixing result for the aforementioned class of functions is obtained via new results: 1) mixing for continuous observables with compact support consisting of configurations at a bounded time from the closest collision; 2) a tightness-type result that allows us to control the configurations with long free flights. To prove 1), we establish a mixing local limit theorem for the Sinai billiard flow with infinite horizon, previously an open question.